We hope you are all staying safe. With massive vaccination programs across the globe we hope you and your loved ones are getting back to what used to be normal. With that out of the way, let us circle back to Property Testing. This month was less sleepy as compared to the two preceding months and we saw six papers in total (two of them explore problems in quantum property testing). Without further ado, let us take a deeper dive.

GSF-locality is not sufficient for proximity-oblivious testing, by Isolde Adler, Noleen Kohler, Pan Peng (arXiv) The notion of proximity oblivious testers was made explicit in the seminal work of Goldreich and Ron in 2009 [GR09]. A proximity oblivious tester for a graph property is a constant query tester that rejects a graph with probability that monotonically increases with distance to the property. (Edit: Correction) A property is called proximity oblivious testable (or PO testable) if it has a one sided proximity oblivious tester. [GR09] gave a characterization of which properties \(\Pi\) are PO testable in the bounded degree model if and only if it is a “local” property of some kind which satisfies a certain non propagation condition. [GR09] conjectured that all such “local” properties satisfy this non propagation condition. This paper refutes the above conjecture from [GR09].

Coming up next. More action on triangle freeness.

Testing Triangle Freeness in the General Model in Graphs with Arboricity \(O(\sqrt n)\), by Reut Levi (arXiv) PTReview readers are likely to be aware that triangle freeness has been a rich source of problems for developing new sublinear time algorithms. This paper considers the classic problem of testing triangle freeness in general graphs. In the dense case, algorithms with running time depending only on \(\varepsilon\) are known thanks to the work of Alon, Fischer, Krivelevich and Szegedy. In the bounded degree case, Goldreich and Ron gave testers with query complexity \(O(1/\varepsilon)\). This paper explores the problem in general graph case and proves an upper bound of \(O(\Gamma/d_{avg} + \Gamma)\) where \(\Gamma\) is the arboricity of the graph. The author also shows that this upperbound is tight for graphs with arboricity at most \(O(\sqrt n)\). Curiously enough, the algorithm does not take arboricity of the graph as an input and yet \(\Gamma\) (the arboricity) shows up in the upper and lower bounds.

Testing Dynamic Environments: Back to Basics, by Yonatan Nakar and Dana Ron (arXiv) Goldreich and Ron introduced the problem of testing “dynamic environments” in 2014. Here is the setup for this problem. You are given an environment that evolves according to a local rule. Your goal is to query some of the states in the system at some point of time and determine if the system is evolving according to some fixed rule or is far from it. In this paper, the authors consider environments defined by elementary cellular automata which evolve according to threshold rules as one of the first steps towards understanding what makes a dynamic environment tested efficiently. The main result proves the following: if your local rules satisfy some conditions, you can use a meta algorithm with query complexity \(poly(1/\varepsilon)\) which is non adaptive and has one sided error. And all the threshold rules indeed satisfy these conditions which means they can be tested efficiently.

Identity testing under label mismatch, by Clement Canonne and Karl Wimmer (arXiv) This paper considers a classic problem distribution testing with the following twist. Let \(q\) denote a distribution supported on \([n]\). You are given access to samples from another distribution \(p\) where \(p = q \circ \pi\) where \(\pi\) is some unknown permutation. Thus, I relabel the data and I give you access to samples from the relabeled dataset. Under this promise, note that identity testing becomes a trivial problem if \(q\) is known to be uniform over \([n]\). The authors develop algorithms for testing and tolerant testing of distributions under this additional promise of \(p\) being a permutation of some known distribution \(q\). The main result shows as exponential gap between the sample complexity of testing and tolerant testing under this promise. In particular, identity testing under the promise of permutation has sample complexity \(\Theta(\log^2 n)\) whereas tolerant identity testing under this promise has sample complexity \(\Theta(n^{1-o(1)})\).

Testing symmetry on quantum computers, by Margarite L. LaBorde and Mark M. Wilde (arXiv) This paper develops algorithms which test symmetries of a quantum states and changes generated by quantum circuits. These tests additionally also quantify how symmetric these states (or channels) are. For testing what are called “Bose states” the paper presents efficient algorithms. The tests for other kinds of symmetry presented in the paper rely on some aid from a quantum prover.

Quantum proofs of proximity, by Marcel Dall’Agnol, Tom Gur, Subhayan Roy Moulik, Justin Thaler (ECCC) The sublinear time (quantum) computation model has been gathering momentum steadily over the past several years. This paper seeks to understand the power of \({\sf QMA}\) proofs of proximity for property testing (recall \({\sf QMA}\) is the quantum analogue of \({\sf NP}\)). On the algorithmic front, the paper develops sufficient conditions for properties to admit efficient \({\sf QMA}\) proofs of proximity. On the complexity front, the paper demonstrates a property which admits an efficient \({\sf QMA}\) proof but does not admit a \({\sf MA}\) or an interactive proof of proximity.

A somewhat “sublinear” month of April, as far as property testing is concerned, with only one paper. (We may have missed some; if so, please let us know in the comments!)

Graph Streaming Lower Bounds for Parameter Estimation and Property Testing via a Streaming XOR Lemma, by Sepehr Assadi and Vishvajeet N (arXiv). This paper establishes space vs. pass trade-offs lower bounds for streaming algorithms, for a variety of graph tasks: that is, of the sort “any \(m\)-pass-streaming algorithm for task \(\mathcal{T}\) must use memory at least \(f(m)\).” The tasks considered include graph property estimation (size of the maximum matching, of the max cut, of the weight of the MST) and property testing for sparse graphs (connectivity, bipartiteness, and cycle-freeness). The authors obtained exponentially improved lower bounds for those, via reductions to a relatively standard problem, (noisy) gap cycle counting, for which they establish their main lower bound. As a key component of their proof, they prove a general direct product result (XOR lemma) for the streaming setting, showing that the advantage for solving the XOR of \(\ell\) copies of a streaming predicate \(f\) decreases exponentially with \(\ell\).

A somewhat quieter month by recent standards. Three Two papers: graph property testing and quantum distribution testing. (Ed: The distribution testing paper was a revision of a paper we already covered in Sept 2020.)

Robust Self-Ordering versus Local Self-Ordering by Oded Goldreich (ECCC). In Nov 2020, we covered a paper that uses a tool called self-ordered graphs, that transferred bit string function lower bounds to graph property testing. Consider a labeled graph. A graph is self-ordered if its automorphism group only contains the identity element (it has no non-trivial isomorphisms). A graph is robustly self-ordered, if every permutation of the vertices leads to a (labeled) graph that is sufficiently “far” according to edit distance. Given a self-ordered graph \(G\), a local self-ordering procedure is the following. Given access to a copy \(G’\) of \(G\) and a vertex \(v \in V(G’)\), this procedure determines the (unique) vertex in \(V(G)\) that corresponds to \(v\) with sublinear queries to \(G\). In other words, it can locally “label” the graph. Intuitively, one would think that more robustly self-ordered graphs will be easier to locally label. This paper studies the relation between robust and local self-ordering. Curiously, this paper refutes the above intuition for bounded-degree graphs, and (weakly) confirms it for dense graphs. Roughly speaking, there are bounded degree graphs that are highly robustly self-ordered, for which any local self-ordering procedure requires \(\omega(\sqrt{n})\) queries. Moreover, there are bounded degree graphs with \(O(\log n)\)-query local self-ordering procedures, yet are not robustly self-ordered even for weak parameters. For dense graphs, the existence of fast non-adaptive local self-ordering procedures implies robust self-ordering.

Testing identity of collections of quantum states: sample complexity analysis by Marco Fanizza, Raffaele Salvia, and Vittorio Giovannetti (arXiv). This paper takes identity testing to the quantum setting. One should think of a \(d\)-dimensional quantum state as a \(d \times d\) density matrix (with some special properties). To learn the state entirely up to error \(\varepsilon\) would require \(O(\varepsilon^{-2} d^2)\) samples/measurements. A recent result of Badescu-O’Donnell-Wright proves that identity testing to a known state can be done significantly faster using \(O(\varepsilon^{-2} d)\) measurements. This paper takes this result a step further by consider a set of \(N\) quantum states. A “sample” is like a classical sample, where one gets a sample from a distribution of quantum states. The YES (“uniform”) case is when all the states are identical. The NO (“far from uniform”) case is when they are “far” from being the same state. This paper proves that \(O(\varepsilon^{-2}\sqrt{N}d)\) samples suffices for distinguishing these cases.

We got quite some action last month. We saw five papers. A lot of action in graph world and some action in quantum property testing which we hope you will find appetizing. Also included is a result on sampling uniformly random graphlets.

Testing Hamiltonicity (and other problems) in Minor-Free Graphs, by Reut Levi and Nadav Shoshan (arXiv). Graph Property Testing has been explored pretty well for dense graphs (and reasonably well for bounded degree graphs). However, testing properties in the general case still remains an elusive goal. This paper makes contributions in this direction and as a first result it gives an algorithm for testing Hamiltonicity in minor free graphs (with two sided error) with running time \(poly(1/\varepsilon)\). Let me begin by pointing out that Hamiltonicity is an irksome property to test in the following senses.

• It is neither monotone nor additive. So the partition oracle based algorithms do not immediately imply a tester (with running time depending only on \(\varepsilon\) for Hamiltonicity. This annoyance bugs you even in the bounded degree case.
• Czumaj and Sohler characterized what graph properties are testable with one-sided error in general planar graphs. In particular, they show a property of general planar graphs is testable iff this property can be reduced to testing for a finite family of finite forbidden subgraphs. Again, Hamiltonicity does not budge to this result.
• There are (concurrent) results by Goldreich and Adler-Kohler which show that with one-sided error, Hamiltonicity cannot be tested with \(o(n)\) queries.

The paper shows that distance to Hamiltonicity can be exactly captured in terms of a certain combinatorial parameter. Thereafter, the paper tries to estimate this parameter after cleaning up the graph a little. This allows them to estimate the distance to Hamiltonicity and thus also implies a tolerant tester (restricted to mino-free graphs).

Testing properties of signed graphs, by Florian Adriaens, Simon Apers (arXiv). Suppose I give you a graph \(G=(V,E)\) where all edges come with a label: which is either “positive” or “negative”. Such signed graphs are used to model various scientific phenomena. Eg, you can use these to model interactions between individuals in social networks into two categories like friendly or antagonistic.

This paper considers property testing problems on signed graphs. The notion of farness from the property extends naturally to these graphs (both in the dense graph model and the bounded degree model). The paper contains explores three problems in both of these models: signed triangle freeness, balance and clusterability. Below I will zoom into the tester for clusterability in the bounded degree setting developed In the paper. A signed graph is considered clusterable if you can partition the vertex set into some number of components such that the edges within any component are all positive and the edges running across components are all negative.

The paper exploits a forbidden subgraph characterization of clusterability which shows that any cycle with exactly one negative edge is a certificate of non-clusterability of \(G\). The tester runs multiple random walks from a handful of start vertices to search for these “bad cycles” by building up on ideas in the seminal work of Goldreich and Ron for testing bipariteness. The authors put all of these ideas together and give a \(\widetilde{O}(\sqrt n)\) time one-sided tester for clusterability in signed graphs.

Local Access to Random Walks, by Amartya Shankha Biswas, Edward Pyne, Ronitt Rubinfeld (arXiv). Suppose I give you a gigantic graph (with bounded degree) which does not fit in your main memory and I want you to solve some computational problem which requires you to solve longish random walks of length \(t\). And lots of them. It would be convenient to not spend \(\Omega(t)\) units of time performing every single walk. Perhaps it would work just as well for you to have an oracle which provides query access to a \(Position(G,s,t)\) oracle which returns the position of a walk from \(s\) at time \(t\) of your choice. Of course, you would want the sequence of vertices returned to behave consistently with some actual random walk sampled from the distribution of random walks starting at \(s\). Question is: Can I build you this primitive? This paper answers this question in affirmative  and shows that for graphs with spectral gap \(\Delta\), this can be achieved with running time \(\widetilde{O}(\sqrt n/\Delta)\) per query. And you get the guarantee that the joint distribution of the vertices you return at queried times is \(1/poly(n)\) close to the uniform distribution over such walks in \(\ell_1\).  Thus, for a random \(d\)-regular graph, you get running times of the order \(\widetilde{O}(\sqrt n)\) per query. The authors also show tightness of this result by showing to get subconstant error in \(\ell_1\), you necessarily need \(\Omega(\sqrt n/\log n)\) queries in expectation.

Efficient and near-optimal algorithms for sampling connected subgraphs, by Marco Bressan (arXiv). As the title suggests, this paper considers efficient algorithms for sampling a uniformly random \(k\)-graphlet from a given graph \(G\) (for \(k \geq 3\)). Recall, a \(k\)-graphlet refers to a collection of \(k\)-vertices which induce a connected graph in \(G\). The algorithm considered in the paper is pretty simple. You just define a Markov Chain \(\mathcal{G}_k\) with all \(k\)-graphlets as its state space. Two states in \(\mathcal{G}_k\) are adjacent iff their intersection is a \((k-1)\)-graphlet. To obtain a uniformly random sample, a classical idea is to just run this Markov Chain and obtain an \(\varepsilon\)-uniform sample. However, the gap between upper and lower bounds on the mixing time of this walk is of the order \(\rho^{k-1}\) where \(\rho = \Delta/\delta\) (that is the ratio of maximum and minimum degrees to the power \(k-1\)). The paper closes this gap up to logarithmic factors and shows that the mixing time of the walk is at most \(t_{mix}(G) \rho^{k-1} \log(n/\varepsilon)\). It also proves an almost matching lower bound. Further, the paper also presents an algorithm with event better running time to return an almost uniform \(k\)-graphlet. This exploits a previous observation: sampling a uniformly random \(k\)-graphlet is equivalent to sampling a uniformly random edge in \(\mathcal{G}_{k-1}\). The paper then proves a lemma which upperbounds the relaxation time of walks in \(\mathcal{G}_k\) to walks in \(\mathcal{G}_{k-1}\). And then you upperbound the mixing time in terms of the relaxation time to get an improved expected running time of the order \(O(t_{mix}(G) \cdot \rho^{k-2} \cdot \log(n/\varepsilon)\).

Toward Instance-Optimal State Certification With Incoherent Measurements, by Sitan Chen, Jerry Li, Ryan O’Donnell (arXiv). The problem of quantum state certification has gathered interest over the last few years. Here is the setup: you are given a quantum state \(\sigma \in \mathbb{C}^{d \times d}\) and you are also given \(N\) copies of an unknown state \(\rho\). You want to distinguish between the following two cases: Does \(\rho = \sigma\) or is \(\sigma\) at least \(\varepsilon\)-far from \(\rho\) in trace norm? Badescu et al showed in a recent work that if entangled measurements are allowed, you can do this with a mere \(O(d/\varepsilon^2)\) copies of \(\rho\). But using entangled states comes with its own share of problems. On the other hand if you disallow entanglement, as Bubeck et al show, you need \(\Omega(d^{3/2}/\varepsilon^2)\) measurements. This paper asks: for which states \(\sigma\) can you improve upon this bound. The work takes inspirations from a la “instance optimal” bounds for identity testing. Authors show a fairly general result which (yet again) confirms that the quantum world is indeed weird. In particular, the main result of the paper implies that the copy complexity of (the quantum analog of) identity testing in the quantum world (with non-adaptive queries) grows as \(\Theta(d^{1/2}/\varepsilon^2)\). That is, the number of quantum measurements you need increases with \(d\) (which is the stark opposite of the behavior you get in the classical world).

Happy 2021! We now cover the last papers of 2020: a (smaller) spread with graph property testing, distribution property testing, and circuit complexity.

Almost Optimal Bounds for Sublinear-Time Sampling of k-Cliques: Sampling Cliques is Harder Than Counting by Talya Eden, Dana Ron, and Will Rosenbaum (arXiv). In many settings, sampling and counting are tightly coupled; the classic example being permanent estimation and sampling random bipartite matchings.This paper investigates the problem of sampling uniform random cliques, and unearths an interesting separation of approximately uniform sampling from approximate counting. Let’s consider the fundamental problem of edge and triangle counting, in graphs of constant arboricity. This class, which contains all minor-closed families, has deep connections to clique counting; clique counting can be done in linear time for bounded arboricity graphs.) The input connected graph \(G\) has \(n\) vertices, \(m\) edges, and \(t\) triangles. We assume the standard query model, with uniform random vertices, neighbor, and degree queries. Our aim is to get \((1+\varepsilon)\)-approximations for \(m\) and \(t\) (in general, any \(k\)-clique count). It is known from previous work that edge estimation can be done in \(O(1)\) time and triangle estimation can be done in \(O(n/t)\) time (ignoring log and \(\varepsilon\) dependencies). Moreover, these bounds is optimal. Can one get similar bounds for approximately sampling a uniform random edge/triangle? Previous work of Eden-Rosenbaum achieve such a bound for sampling edges. Surprisingly, this paper shows that triangle sampling is fundamentally harder, and requires \(\Omega(n^{3/4}/\sqrt{t})\) queries. The main result gives a precise and optimal bound for uniform sampling \(k\)-cliques in graphs of arboricity \(\alpha\).

Testing and reconstruction via decision trees by Guy Blanc, Jane Lange, and Li-Yang Tan (arXiv). A property tester is an (sublinear) algorithm that distinguishes an object with a property from those that are far. Reconstruction is a much stronger sublinear algorithm, which actually provides query access to the closest object. Suppose we have query access to an \(n\)-variate Boolean function \(f\), and the property \(\mathcal{P}\) of interest is size-\(s\) decision trees. A reconstructor would give query access to a function \(g \in \mathcal{P}\), such that \(dist(f,g) = O(OPT_f)\), where \(OPT_f\) is the distance of \(f\) to \(\mathcal{P}\). Observe that a reconstructor automatically gives a distance estimator (just estimate \(dist(f,g)\) by random sampling). One of the main results of this paper is a (weaker, bicriteria) reconstructor, where \(g\) is guaranteed to have decision-tree complexity \(s^{O(\log^2s)}\). Each query to \(g\) is computed in \(poly(\log s, 1/\varepsilon)\cdot n\) time. This reconstructor leads to a number of testing results, and reconstructors for other properties like low Fourier degree. Another interesting result proven: if there exists an efficient reconstructor that gets \(g\) of size \(s\), then there exists an efficient proper learner for decision trees (a big open problem).

Testing Product Distributions: A Closer Look by Arnab Bhattacharyya, Sutanu Gayen, Saravanan Kandasamy, and N. V. Vinodchandran (arXiv). This paper is on distribution testing, where all distributions involved are promised to be product distributions. Let \(P, Q\) be distributions over \(\Sigma^n\), where \(\Sigma\) is some (small) finite set. The problems studied are identity testing (\(Q\) is known/fixed, and one gets samples from unknown \(P\)) and closeness testing (both unknown). It is known from previous work that these problems can basically be solved with \(O(n|\Sigma|\varepsilon^{-2})\) samples. Note that the domain size is exponential in \(n\); thus, these bounds give an exponential improvement over identity/closeness testing in the vanilla (non-product) setting. Tolerant testing is of special significance here. Because the domain is so large, it makes sense to allow for some distance even in the “yes” case. The main result of this paper is to completely map out the complexities for identity/closeness testing where we wish to distinguish \(d_1(P,Q) < \varepsilon_1\) from \(d_2(P,Q) > \varepsilon_2\), where \(d_1, d_2\) are potentially different distance measures. For example, the paper considers total variation (TV), KL, Hellinger, and \(\chi^2\) distances. An important improvement achieved is for identity testing where \(d_1\) is Hellinger and \(d_2\) is TV distance: for certain values of \(\varepsilon_1, \varepsilon_2\), one can get a tester of optimal sample complexity \(O(\sqrt{n|\Sigma|})\). Previous bounds only gave a \(O(n|\Sigma|)\) sample complexity. There are a number of new lower bounds for different combinations of \(d_1, d_2\) distance pairs.

Highlights in property testing this month, include developments across the board. We have a nice potpourri of results which will catch your fancy. From a result which shows the tightness of quadratic gap between query complexities of adaptive vs non-adaptive testers for dense graph properties, to isoperimetric Talagrand-like inequalities for real valued functions over the hypercube, to results which consider face off between quantum computing and distribution testing and more. Looks like the festive season came early for the PTReview readers.

Non-adaptive vs Adaptive Queries in the Dense Graph Testing Model (by Oded Goldreich and Avi Wigderson) (ECCC) It has been long established in the dense graph property testing literature that the query complexity of adaptive testers and non-adaptive testers are only a quadratic factor apart. This work asks: is this gap necessarily tight and answers in the affirmative. The main conceptual tool used in the paper is the notion of robustly self-ordered graphs and local self-ordering procedures for these graphs (which was developed by the same authors and was covered in our September Post). These results use these tools to port lower bounds on property testing problems for bit strings to graph properties.

Direct Sum and Partitionability Testing over General Groups (by Andrej Bogdanov and Gautam Prakriya) (ECCC) In their celebrated work, Blum, Luby and Rubinfeld gave a four query test to determine whether a function \(f \colon \mathbb{F}_2^n \to \mathbb{F}_2\) is affine. This paper considers a generalization of the notion of affinity to functions \(f \colon \{0,1\}^n \to G\) where \(G\) is an Abelian group. The generalization sough asks: does \(f\) have the form \(f = \sum x_ig_i + g_0\) for group elements \(g_0, g_1, \cdots, g_n \in G\). In this setting, the BLR analysis does not apply as the domain and range are note equipped with the same group structure. This work presents a four query test for the above problem. In fact, the test is more general and can be used to decide whether a function \(f \colon D_1 \times D_2 \ldots \times D_n \to G\) from a product domain \(D_1 \times D_2 \ldots \times D_n\) to an Abelian group $G$ is a direct sum if it has the form \(f(x_1, x_2, \cdots, x_n) = \sum f_i(x_i)\) which resolves a conjecture of Dinur and Golubev. The work also presents results for testing whether a function \(f\) defined from a product domain to an Abelian group \(G\) is a direct product.

Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing (by Hadley Black, Iden Kalemaj and Sofya Raskhodnikova) (ArXiv) This paper generalizes the celebrated isoperimetric inequalities of boolean functions over the hypercube to the setting of real valued functions. This generalized isoperimetry is then put to work to obtain a monotonicity tester for \(f \colon \{0,1\}^n \to \mathbb{R}\). The tester makes \(\min(\tilde{O}(r \sqrt n), O(n))\) non-adaptive queries (where \(r\) is the size of image of \(f\)) and has one-sided error. The authors generalize the KMS inequality by a Boolean decomposition theorem which allows them to represent any real valued function over the cube as a collection of Boolean functions over the cube which capture \(f\)’s distance to montonicity as well the structure of violations to monotonicity in \(f\).

Erasure-Resilient Sublinear-Time Graph Algorithms (by Amit Levi, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova and Nithin Varma) (ArXiv) In a previous work, a subset of the authors explored how to equip property testers with the erasure-resilience for function properties. With graph properties, a different picture emerges. In the dense graph model, where you have query access to the adjacency matrix, the situation is still fine: adjacency matrices are functional representations of graphs: therefore, if you have a black belt in making property testers erasure resilient for function properties, you would be able to test properties of dense graphs too. However, if I give you query access to the graph through an adjacency list, the picture changes. These are non-functional representations of graphs and therefore need new conceptual tools. This paper begins the study of erasure resilience in the adjacency list model. It focuses on two computational tasks: testing connectivity and estimating average degree. Let me showcase the results in the paper on testing connectivity. It is shown that you encounter a threshold phenomena in testing connectivity. As long as the fraction of erasures is small, you get testers which run in time independent of the size of the graph. But when the fraction of erasures exceeds a certain cutoff, it is shown that the tester needs a number of queries linear in the size of the adjacency list.

Expander Random Walks: A Fourier Analytic Approach (by Gil Cohen, Noam Peri and Amnon Ta-Shma) (ECCC) While this is a not exactly a property testing paper, how can you not report a paper which proves a significant strengthening of the classic CLT for Markov Chains (alright, for random walks on expanders) with respect to the TV distance? Let us now unpack the above a little. So, consider the following setup: Suppose you have a \(d\)-regular expander \(G\). Now, imagine running the following process \(\textsf{(P)}\):

1) Label half the vertices in \(G\) as \(+1\) and the other half as \(-1\) arbitrarily.
2) Take a \(t-1\) step random walk which involves \(t\) vertices: say \(v_1, v_2, \ldots v_t\).
3) Finally, return the boolean label of all of the visited vertices. This gives a bit-vector \(x \in \{-1,1\}^t\).

Now, let us ask: Which class of functions get fooled by this process?

The main result of the paper shows that this process fools

1) Symmetric functions
2) The class \(AC^0\)
3) Functions \(f\) which are computable by low width read once branching programs.

In more detail, let \(f\) be a symmetric function and let \(G\) be a \(\lambda\)-spectral expander. For the process \(\textsf{(P)}\) defined above, it follows by the spectral expansion of \(G\), that \(discrepancy(f) = |E_{x \sim \textsf{(P)}} [f(x)] – E_{x \sim U_t} [f(x)]|\) is small.

Now note that the total variation distance is precisely equal to the maximum discrepancy you get over all symmetric functions \(f\). This leads to the connection that allows the authors to upgrade CLT for expander random walks as they are able to show a CLT for Markov Chains with respect to the TV distance (as opposed to the CLT with respect to the Kolmogorov Distance which is what was known from the work of Kipnis and Vadhan since 1986).

New Sublinear Algorithms and Lower Bounds for LIS Estimation (by Ilan Newman and Nithin Varma) (ArXiv) As the title suggests, this paper considers the task of estimating the length of the longest increasing sequence in an array. In particular, it gives a non-adaptive algorithm which estimates the LIS length within an additive error of \(\pm \varepsilon n\) in \(\tilde{O}\left(\frac{r}{\varepsilon^3}\right)\) queries (where \(r\) is the number of distinct elements in \(A\)). On the lower bound side, the paper presents a lower bound of \((\log n)^{\Omega(1/\varepsilon)}\) non-adaptive queries. The lower bound construction uses ideas from lower bound of Ben-Eliezer, Canonne, Letzter and Waingarten on testing monotone pattern freeness.

Stability and testability: equations in permutations (by Oren Becker, Alexander Lubotzky, AND Jonathan Mosheiff) (ArXiv) Consider the following question: Suppose I ask if two permuations \(A,B \in Sym(n)\). Suppose I want to decide whether these permutations commute or whether they are far from permutations which commute. Can this question be answered in time independent of \(n\)? The authors answer this question in the affirmative. They show a simple Sample and Substitute procedure which essentially does the following: take samples \(x_1, x_2, cdots x_k \sim [n]\). And accept iff \(ABx_i = BAx_i\) for each \(i\). The authors call this particular set of equations/constraints (checking whether \(AB = BA\)) stable: as it admits a Sample and Substitute style tester with query complexity independent of \([n]\). This allows the authors to consider the group theoretic notion of stability as a property testing concept and allows them to examine the notion of stability from a computational standpoint.

StoqMA meets distribution testing (by Yupan Liu) (ArXiv) This paper shows a containment result. A certain complexity class \(\textsf{eStoqMA}\) is shown to be a subset of \(\textsf{StoqMA}\). Let us informally define what these classes are. A language \(L \in \textsf{StoqMA}\) if there exists a verifier (which is a reversible quantum circuit) such that on input a string \(x\) such that for every \(x \in L\) there exists a witness quantum state which makes the verifier accept with probability at least \(2/3\). And in case \(x \not\in L\), for every quantum state the verifier rejects \(x\) with probability at least \(2/3\). The paper characterizes this class from a distribution testing point of view. And this connection allows the author to conclude that \(\textsf{eStoqMA} = \textsf{MA}\). Here, \(\textsf{eStoqMA}\) is a sub-class of \(\textsf{StoqMA}\) where all YES instances have an easy witness.

(Edit: Added later) We missed the following two papers. Thanks to our readers for pointing it out.

Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu (by Arnab Bhattacharya, Sutanu Gayen, Eric Price and N.V. Vinodchandran) (ArXiv) Graphical models are a convenient way to describe high dimensional data in terms of its dependence structure. An important question in this area concerns learning/inferring the underlying graphical model from iid samples. This paper considers the problem when the underlying graph is in fact a tree. Thus, the challenge is to learn a tree structured distribution. The main result is that given sample access to a tree structured distribution, you can recover an \(\varepsilon\) approximate tree with good probability in \(O(n/\varepsilon)\).

Relaxed Locally Correctable Codes with Improved Parameters (by Vahid R. Asadi and Igor Shinkar) (ECCC) In their work on Robust PCPs of Proximity, Ben-Sasson et al. asked whether it is possible to obtain a \(q\)-query (Relaxed) Locally Decodable Code whose block length is strictly smaller than the best known bounds on block lengths for Locally Decodable Codes. Recall with relaxed LDCs you are allowed to abort outputting the \(i^{th}\) bit of the message should you detect that the received word is not a valid codeword. This paper makes progress on this problem and shows that you can actually construct an \(O(q)\)-query relaxed LDCs which encode a message of length \(k\) using \(O(k^{1 + 1/q})\) queries. This matches some of the known lower bounds for constant query LDCs which thus makes progress towards understanding the gap between LDCs and relaxed LDCs in the regime \(q = O(1)\).

Apologies dear readers for the late posting. The beginning of the school year is always frenzied, and the pandemic has only added to that frenzy. We have an exciting September, with four papers on graph property testing, one two papers on distribution testing, and one paper that connects both topics.

(Ed: we normally scan through ECCC and arXiv, but are happy to post about papers that appear elsewhere. Thanks to the reader who pointed out a relevant COLT 2020 paper.)

Estimation of Graph Isomorphism Distance in the Query World by Sourav Chakraborty, Arijit Ghosh, Gopinath Mishra, and Sayantan Sen (ECCC). Graph isomorphism is about as fundamental as it gets, and this papers studies approximating the graph isomorphism distance for dense graphs. There is a known graph \(G_k\) (with \(n\) vertices). The algorithm is given query access to an input graph \(G_u\) and needs to approximate the number of edge inserts/deletes required to make the graphs isomorphic. This is the tolerant testing version; the property testing version is known to be doable in \(\widetilde{O}(\sqrt{n})\) queries (Matsliah-Fischer). The main insight of this paper is to relate the tolerant testing complexity to a distribution testing problem. Consider distributions over the \(\{0,1\}^n\) defined by multisets of \(n\) hypercube points. Our aim is to estimate the earthmover distance between a known distribution and an unknown distribution. Interestingly, the query model is different: one can sample the underlying multisets without replacement. It turns out that the optimal complexity of this problem is (upto polylog factors) is the same as the optimal complexity of tolerant testing of graph isomorphism. A direct corollary is that the isomorphism distance can be approximated upto additive \(\epsilon n^2\) using \(\widetilde{O}(n)\) samples. This equivalence also gives an alternate proof for lower bounds for property testing graph isomorphism.

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing by Oded Goldreich and Avi Wigderson (ECCC). Let’s start from the application. The aim is to prove the following property testing lower bounds for the bounded-degree graph setting: an exponential separation between tolerant and vanilla testing, and finding an efficiently decidable property (in polynomial time) that cannot be property tested in sublinear time. For binary strings, results of this form are known. Can these be “ported” to the bounded-degree graph world? Can we construct graphs such that adjacency queries reduce to bit queries in strings? Naturally, one can simply represent the adjacency list as a string and treat graph queries as bit queries. But the problem is that of isomorphisms: different bit strings could represent the same graph and therefore, the different bit strings must have the same status with respect to the underlying property. The key insight in this paper is to introduce robustly self-ordered graphs, as a tool to port bit string property testing lower bounds to bounded-degree graphs. Such graphs essentially have a unique (identity) automorphism, even after a few edge insert/deletes. The actual definition is more technical, but that is the essence. The main result is an explicit construction of such graphs, from which the lower bound can be ported directly through a convenient lemma.

Modifying a Graph’s Degree Sequence and the Testablity of Degree Sequence Properties by Lior Gishboliner (arXiv). A sequence of numbers \(D = (d_1, d_2, \ldots, d_n)\) is graphic if there exists an undirected graph on \(n\) vertices whose degrees are precisely the numbers of the sequence. Graphical sequences have been characterized by classic results of Erdös-Gállai and Havel-Hakimi. This paper first proves the following theorem. Suppose a graphic sequence \(D’\) has \(l_1\)-distance at most \(\delta\) from the degree sequence \(D\) of a graph \(G\). Then, there exists a graph \(G’\) with degree sequence \(D’\) such that the (dense graph) distance between \(G\) and \(G’\) is \(O(\sqrt{\delta})\). This theorem is used to prove an interesting property testing result. Let \(\mathcal{D}\) be a subset of graphic sequences that are closed under permutation. Let \(\mathcal{G}\) be the set of graphs that have a degree sequence in \(\mathcal{D}\). Then \(\mathcal{G}\) can be tested in \(poly(1/\epsilon)\) queries.

Sampling an Edge Uniformly in Sublinear Time by Jakub Têtek (arXiv). In the general model for sublinear algorithms on graphs, an important choice is whether one allows uniform random edge queries. A natural question is whether such queries can simulated efficiently, using only random vertex, degree, and neighbor queries. This problem appears somewhat implicitly in previous sublinear subgraph counting algorithms, and Eden-Ron-Rosenbaum study it explicitly. They prove that one can sample from an \(\epsilon\)-approximate uniform distribution (over edges) using \(O(n/\sqrt{\epsilon m})\) samples. The problem of sampling from exactly the uniform distribution is left open. Until this paper. The main result shows that by modifying the Eden-Ron-Rosenbaum algorithm parameters, one can generate edge samples from an \(\epsilon\)-approximate uniform distribution using \(O((n/\sqrt{m})\log \epsilon^{-1})\) samples. The exact uniform distribution is achieved by setting \(\epsilon = 1/n\), to get a sample complexity of \(O((n\log n)/\sqrt{m})\).

Faster Property Testers in a Variation of the Bounded Degree Model by Isolde Adler and Polly Fahey (arXiv). The setting of bounded-degree graph property testing naturally extends to bounded-degree relational databases, which can be thought of as “directed” hypergraphs. This is an interesting new direction of research, that combines property testing with database theory (see Adler-Harwath and Chen-Yoshida). One of the main contributions of this work is to consider another notion of distance: edge and vertex inserts/deletes. This is a natural extension, and we can now compare distances between graphs/databases with different numbers of vertices. The main result is that, under this notion of distance, a large class of properties can be tested in constant running time on databases with bounded degree and treewidth. Specifically, any property expressible in Counting Monadic Second-Order Logic (CMSO) can be tested in constant time. Previous results by Alder-Harwath showed that such properties can be tested (under the standard distance notion) in constant queries, but polylogarithmic time.

Optimal Testing of Discrete Distributions with High Probability by Ilias Diakonikolas, Themis Gouleakis, Daniel M. Kane, John Peebles, and Eric Price (arXiv, ECCC). The focus of this paper is distribution testing in the “high probability” regime, where we wish the error of the tester to be \(< \delta\). Typically, most results just get an error of at most \(1/3\), from which standard probabilistic boosting would tack on an extra \(O(\log 1/\delta)\) factor. In standard TCS settings, one doesn’t focus on optimizing this dependence, but in statistics, there is significant focus on the optimal sample complexity. And indeed, for practical applications, it is crucial to have sharp bounds on the right number of samples required for hypothesis testing. The paper also argues that getting the optimal sample complexity requires new algorithms, even for uniformity testing. There are optimal results given for closeness and independence testing. The optimal sample complexity only pays a multiplicative factor of \(\log^{1/3} (1/\delta)\) or \(\log^{1/2}(1/\delta)\) over the optimal bound for constant error (with other additive terms depending on \(\log(1/\delta)\)).

Bessel Smoothing and Multi-Distribution Property Estimation by Yi Hao and Ping Li (COLT 2020). Let us consider some standard (tolerant) distribution testing questions, phrases as approximation algorithms. Given sample access to two distributions \(p\) and \(q\) over \([n]\), we may wish to estimate the \(l_1\)-distance, \(l_2\)-distance, relative entropy, etc. between these distributions. One can phrases this problem abstractly as estimating \(\sum_{i \in [n]} f(p_i, q_i)\), where \(f\) is some explicit function. This papers shows that for any 1-Lipschitz function \(f\) that satisfies some “regularity” property, the sum \(\sum_{i \in [n]} f(p_i, q_i)\) can be \(\epsilon\)-approximated with \(O(\epsilon^{-3}n/\sqrt{\log n})\) samples (apologies to the authors to replacing their \(k\) with the more familiar \(n\) for our readers). Thus, we can get sublinear sampling complexity for a very general class of estimation problems. Moreover, this was actually the simplest setting consider in the paper. One can deal with such functions of \(d\) distributions, not just two distributions. One of the corollaries of the theorems is a sublinear tolerant tester for the property of being a mixture of distributions.

Last month saw action in property testing across the board: graphs, distributions and functions were all objects of study in papers that came out last month. Also included is a separation result between quantum and classical query complexities resolving a conjecture of Aaronson and Ambainis. Here is a brief report.

Testing asymmetry in bounded degree graphs, by Oded Goldreich (ECCC). This paper studies a natural graph property hitherto not considered in the property testing literature. Namely, the question of testing whether a graph is asymmetric or whether it is far from being asymmetric. A graph is said to be asymmetric if its automorphism group is trivial (that is, it only contains the identity permutation). One of the results in the paper says that this problem is easy in the dense graph model – which is a side result of the paper. This is because all dense graphs are \(O(\log n/n)\)-close to being asymmetric. To see this, the paper points out that a simple randomized process which takes \(G\) as input and returns an asymmetric graph by changing very few edges. This process asks you to do the following: Take a set \(S \subseteq V\) with \(|S| = O(\log n)\) nodes and replace the subgraph they induce with a random graph. Moreover,  randomize all the edges between \(S\) and \(V \setminus S\). What you can show is that in this modified graph, any automorphism (whp) will map \(S\) to itself. And all the remaining vertices behave (whp) in a unique manner which is peculiar to it. In particular, this means that any automorphism better not map a vertex \(v\) in \(V \setminus S\) to any other vertex in \(V \setminus S\). And this finishes the argument. The main result explores the bounded degree model. By a simple argument, you can show that testing asymmetry is easy if all the connected components have size \(s(n) \leq o\left(\frac{\log n}{\log {\log n}}\right)\). Thus, the challenging case is when you have some connected components of a larger size. In this case, what Goldreich shows is the following: If all the components have size at most \(s(n)\) where \(s(n) \geq \Omega\left(\frac{\log n}{\log {\log n}}\right)\), then you can test asymmetry (with one-sided-error) in \(O(n^{1/2} \cdot s(n)/\epsilon)\) queries.  Moreover, the paper also shows a two-sided-lower bound of \(\Omega\left(n/s(n)\right)^{1/2}\) queries which holds as long as \(\epsilon \leq O(1/s(n)\). This leaves open the bounded degree question of determining the query complexity of testing asymmetry in the general case as the paper also points out.

On testability of first order properties in Bounded degree graphs, by Isolde Adler, Noleen Köhler and Pan Peng (arXiv). One of the well understood motivations for property testing begins in the following manner. Decision problems are typically hard to solve if they involve a universal quantifier — either \(\exists\) or \(\forall\). One way around this hardness is to do the following ritual: relax the problem by dropping the universal quantifier, define a notion of distance between objects in your universe and ask a promise problem instead. Indeed, if you take your favorite property testing problem, you will note that it precisely fits in the template above. How about making this business more rigorous in bounded degree graph property model? This is precisely the content of this work which considers the face off between (First Order) logic and property testing in the bounded degree model. The authors show some interesting results. They begin by showing that in the bounded degree model, you can show that all first order graph properties which involve a single quantifier \(Q \in \{\forall, \exists\}\) are testable with constant query complexity.
If you find this baffling, it would be good to remind yourself that not all graph properties can be expressed in the language of first order logic with a single quantifier! So, you can rest easy. The graph properties you know are not constant time testable are most assuredly not expressible with a single quantifier in First Order Logic. However, this work shows more. It turns out, that “any FO property that is defined by a formula with quantifier prefix \(\exists^* \forall^*\) is testable”. Moreover, there do exist FO graph properties defined by the quantifier prefix \(\forall^* \exists^*\) which are not testable. Thus, this work achieves results in the bounded degree model which are kind of analogous to the results in the dense graph model by Alon et al [1]. On a final note, I find the following one sentence summary of the techniques used to prove the lower bound rather intriguing: “[the paper] obtains the lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs.”

On graphs of bounded degree that are far from being Hamiltonian, by Isolde Adler and Noleen Köhler (arXiv). This paper explores the question of testing Hamiltonicity in the bounded degree model. The main result of the paper is that Hamiltonicity is not testable with one-sided error with \(o(n)\) queries. PTReview readers might recall from our July Post a concurrent paper by Goldriech [2] which achieves the same lower bound on query complexity in the two-sided error model (the authors call attention to [2] as well). One of the interesting feature of this result is that the lower bounds are obtained by an explicit deterministic reduction as opposed to the usual randomized reduction. Like the authors point out, this offers more insights into structural complexity of instances that are far from being Hamiltonian. We point out that this also differs from how the lower bound is derived in [2] — which is via local hardness reductions to a promise problem of 3 CNF satisfiability.

An optimal tester for \(k\)-linear, by Nader Bshouty (ECCC). This paper explores two related questions. We call a function \(f \colon \{0,1\}^n \to \{0,1\}\) \(k\)-linear if it equals the \(\sum_{i \in S} x_i\) for some \(S \subseteq [n]\) of size exactly \(k\). A boolean function is said to be \(k\)-linear* if it is \(j\) linear for a fixed \(j\) where \(j \in \{0,1,2, \cdots, k\}\). The paper proves the following theorems.

1. There exists a non-adaptive one-sided distribution free tester for \(k\)-linear* with query complexity being \(O\left(k \log k + \frac{1}{\varepsilon}\right)\). This matches the two-sided lower bound (where the underlying distribution is uniform) by Blais et al [3].
2. Using a reduction from \(k\)-linear* to \(k\)-linear, the paper shows one can obtain a non-adpative two-sided distribution free tester for \(k\)-linear with same query complexity as the above result. The lower bound from Blais et al applies here also (in fact, they prove a lower bound on \(k\)-linearity).
3. Next up, the paper has a couple of lower bound results to accompany this. One of these results reveals the price you pay for being one-sided and exact (that is, you insist on the function being exactly \(k\)-linear). Turns out, now you have a non-adaptive one-sided uniform distribution lower bound of \(\widetilde{\Omega}(k) \log n + \Omega(1/\varepsilon)\). If you allow adaptivity instead, the paper shows a lower bound of \(\widetilde{\Omega}(\sqrt k)\log n + \Omega(1/\varepsilon)\).

Amortized Edge Sampling, by Talya Eden, Saleet Mossel and Ronitt Rubinfeld (arXiv). Consider the following setup. You are given query access to adjacency list of a graph \(G\) with \(n\) vertices and \(m\) edges. You can make degree queries and neighbor queries. Suppose I ask you to sample a single edge from this graph from a distribution that is pointwise \(\varepsilon\) close to the uniform distribution. Eden and Rosenbaum already showed how you can achieve this with a budget of \(\widetilde{\Theta}(n/\sqrt m)\) queries. Starting from this jump off point, the authors ask whether you can circumvent this lower bound if you want to return multiple samples from a distribution which is again pointwise close to uniform. The paper answers this question in the affirmative and shows that if you know the number of samples, \(q\), in advance you can get away with an amortized bound of \(O*(\sqrt q n/\sqrt m)\) on the total number of queries needed.

On the High Accuracy Limitation of Adaptive Property Estimation, by Yanjun Han (arXiv). Take a discrete distribution \(\mathcal{P}\) with support size \(k\) and consider the task of estimating some symmetric property of \(\mathcal{P}\) to a small \(\pm \varepsilon\) additive error. Here, a symmetric property refers to a “nice” functional defined over the probability simplex, i.e., it refers to functions \(F \colon \Delta_k \to \mathbb{R}\) where \(F(p) = \sum_{i=1}^{k} f(p_i)\) where \(f \colon (0,1) \to \mathbb{R}\). A naive attack to these estimation tasks goes through the following ritual: you get your hands on the empirical distribution, you plug it in \(F\) and you hope for the best. Turns out, you are out of luck if the function \(f\) is non-smooth and in these cases you end up with a suboptimal estimator. Previous works have also looked at more sophisticated estimators (like the local moment matching or LMM and profile maximum likelihood or PML estimator). Turns out, using the LMM or PML estimator leads to optimal sample complexity for a handful of symmetric properties (as long as \(\varepsilon \geq n^{-1/3}\)). This paper considers the question of what can you say for supersmall values of \(\varepsilon\) where \(n^{-1/2} \leq \varepsilon \leq n^{-1/3}\). (The \(n^{-1/2}\) appears because there are estimators that use the knowledge of \(f\) and \(\varepsilon\) can be driven all the way down to \(n^{-1/2}\) for these estimators). The paper focuses on estimators which do not exploit any structure in \(f\). In particular, the paper specializes this question to PML and shows a fundamental limitation on PML which means that the PML approach fails to be sample optimal for the entire range of \(\varepsilon\) and is sample optimal only for \(\varepsilon >> n^{-1/3}\) — which also confirms a conjecture of Han and Shiragur (and refutes a conjecture of Acharya et al. who postulated this is sample optimal for the entire range of \(\varepsilon\)).

\(k\)-Forrelation Optimally Separates Quantum and Classical Query
Complexity, by Nikhil Bansal and Makrand Sinha (arXiv). Understanding the power of quantum query over classical queries is a well motivated problem with a rich history. One of the biggest questions in this area asks for the largest separation between classical and quantum query complexities. In a breakthrough, Aaronson and Ambainis [4] showed a fundamental simulation result which confirmed that you can simulate \(q\) quantum queries with \(O(N^{1 – 1/2q})\) classical queries in the randomized decision tree model of computation as long as \(q = O(1)\). In the same paper, the authors also showed that the standard forrelation* problem exhibits a \(1\) versus \(\widetilde{\Omega}(\sqrt n)\) separation. This means that for \(q = 1\), you essentially have optimal separation. But what about \(q > 1\)? To this end, Aaronson and Ambainis conjectured that a certain problem which they called \(k\)-forrelation — which can be computed with \(q = k/2\) queries requires at least \(\widetilde{\Omega}(n^{1-1/k})\) classical queries. The current work precisely confirms this conjecture.

(*) The forrelation problem asks you to decide whether one Boolean function is highly correlated with the Fourier transform of a second function.

(Edit: Added Later) Simon Apers points out a paper by Shrestov, Storozhenko and Wu that we missed. (Thanks Simon)! Here is a brief report on that paper.

An optimal separation of randomized and quantum query complexity (by Alexander Shrestov, Andrey Storozhenko and Pei Wu)(arXiv) Similar to the paper by Bansal and Sinha [BS20] mentioned above, this paper also resolves the conjecture by Aaronson and Ambainis proving the same result. Like the paper also notes, the techniques in both of these works are completely different and incomparable. On the one hand [BS20] proves the separation for an explicit function as opposed to a function chosen uniformly at random from a certain set as considered in this work. On the other hand, the separation result shown in [BS20] only applies when the query algorithm returns the correct answer with probability at least \(1/2 + 1/poly(\log n)\) — in contrast the results in this paper apply even when the query algorithm is required to have probability of correctness be a constant at least \(1/2\). In addition, this work also proves the \(\ell\)-Fourier weight conjecture of Tal which is of independent interest beyond quantum computing.

So, it looks like all in all we had a great month with two concurrent papers both resolving Aaronson Ambainis conjecture (yet again after two concurrent papers on testing Hamiltonicity)!

References:

[1] Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of
large graphs. Combinatorica, 20(4):451–476, 2000.

[2] Oded Goldreich. On testing hamiltonicity in the bounded degree graph model. Electronic Colloquium on Computational Complexity (ECCC), (18), 2020

[3] Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. CCC 2011

[4] Scott Aaronson and Andris Ambainis. Forrelation: A problem that optimally separates quantum from classical computing. SIAM J. Comput., 47(3):982–1038, 2018

We hope you’re all staying safe and healthy! To bring you some news (and distraction?) during this… atypical summer,here are the recent papers on property testing and sublinear algorithms we saw appear this month. Graphs, probability distributions, functions… there is a something for everyone.

On Testing Hamiltonicity in the Bounded Degree Graph Model, by Oded Goldreich (ECCC). The title sort of gives it away: this relatively short paper shows that testing whether an unknown bounded-degree graph has a Hamiltonian path (or Hamiltonian cycle) in the bounded-degree model requires a number of queries linear in \(n\), the number of nodes. The results also hold for directed graphs (with respect to directed Hamiltonian path or cycle), and are shown via a local reduction to a promise problem of satisfiability of 3CNF formulae. Also included: a complete proof of the linear lower bound for another problem, Independent Set Size; and an open problem: what is the query complexity of testing graph isomorphism in the bounded-degree model?

Local Access to Sparse Connected Subgraphs Via Edge Sampling, by Rogers Epstein (arXiv). Given access to a connected graph \(G=(V,E)\), can we efficiently provide access to some sparse connected subgraph \(G’=(V,E’)\subseteq G\) with \(|E’|\ll |E|\)? This question, well-studied in particular for the case where \(G\) had bounded degree and the goal is to achieve \(|E’|\leq (1-\varepsilon)|V|\), is the focus of this paper which provides a trade-off between the query complexity of the oracle and \(|E’|\). Specifically, for every parameter \(T\), one can give oracle access to \(G’\) with \(|E’|=O(|V|T)\), with a query complexity \(=\tilde{O}(|E|/T)\).

Switching gears, we move from graphs to probability distributions:

Tolerant Distribution Testing in the Conditional Sampling Model, by Shyam Narayanan (arXiv). In the conditional sampling model for distribution testing, which we have covered a few times on this blog, the algorithm at each step gets to specify a subset \(S\) of the domain, and observe a sample from the distribution conditioned on \(S\). As it turns out, this can speed things up a lot: as Canonne, Ron, and Servedio (2015) showed, even tolerant uniformity testing, which with i.i.d. samples requires a near-linear (in the domain size \(n\)) number of samples, can be done in a constant number of conditional queries. Well, sort of constant: no dependence on \(n\), but the dependence on the distance parameter \(\varepsilon\) was, in CRS15, quite bad: \(\tilde{O}(1/\varepsilon^{20})\). This work gets rid of this badness, and shows the (nearly) optimal \(\tilde{O}(1/\varepsilon^{2})\) query complexity! Among other results, it also generalizes it to tolerant identity testing (\(\tilde{O}(1/\varepsilon^{4})\)), for which previously no constant-query upper bound was known. Things have become truly sublinear.

Interactive Inference under Information Constraints, by Jayadev Acharya, Clément Canonne, Yuhan Liu, Ziteng Sun, and Himanshu Tyagi (arXiv). Say you want to do uniformity/identity testing (or learn, but let’s focus on testing) on a discrete distribution, but you can’t actually observe the i.i.d. samples: instead, you can only do some sort of limited, “local” measurement on each sample. How hard is the task, compared to what you’d do if you fully had the samples? This setting, which captures things like distributed testing with communication or local privacy constraints, erasure channels, etc., was well-understood from previous recent work in the non-adaptive setting. But what if the “measurements” could be made adaptively? This paper shows general lower bounds for identity testing and learning, as a function of the type of local measurement allowed: as a corollary, this gives tight bounds for communication constraints and local privacy, and shows the first separation between adaptive and non-adaptive uniformity testing, for a type of “leaky” membership query measurement.

Efficient Parameter Estimation of Truncated Boolean Product Distributions, by Dimitris Fotakis, Alkis Kalavasis, and Christos Tzamos (arXiv). Suppose there is a fixed and unknown subset \(S\) of the hypercube, a “truncation” set, which you can only accessible via membership query; and you receive i.i.d. samples from an unknown product distribution on the hypercube, truncated on that set \(S\) (for instance, because your polling strategy or experimental measurements have limitations). Can you still learn that distribution efficiently? Can you test it for various properties, as you typically really would like to? (or is it just me?) This paper identifies some natural sufficient condition on \(S\), which they call fatness, under which the answer is a resounding yes. Specifically, if \(S\) satisfies this condition, one can actually generate honest-to-goodness i.i.d. samples (non-truncated) from the true distribution, given truncated samples!

Leaving distribution testing, our last paper is on testing functions in the distribution-free model:

Downsampling for Testing and Learning in Product Distributions, by Nathaniel Harms and Yuichi Yoshida (arXiv). Suppose you want to test (or learn) a class of Boolean functions \(\mathcal{C}\) over some domain \(\Omega^n\), with respect to some (unknown) product distribution (i.e., in the distribution-free testing model, or PAC-learning model). This paper develops a general technique, downsampling, which allows one to reduce such distribution-free testing of \(\mathcal{C}\) under a product distribution to testing \(\mathcal{C}\) over \([r]^d\) under the uniform distribution, for a suitable parameter \(r=r(d,\varepsilon,\mathcal{C})\). This allows the authors, among many other things and learning results, to easily re-establish (and, in the second case, improve upon) recent results on testing of monotonicity over \([n]^d\) (uniform distribution) and over \(\mathbb{R}^d\) (distribution-free).

Sublinear algorithms in times of social distancing…always something exciting. This month we have a slew of results on sublinear algorithms for classic graph problems.

(Ed: We have removed a previously posted paper due to correctness concerns raised by our readers. Please look at the post on our paper policy.)

Palette Sparsification Beyond (∆ + 1) Vertex Coloring by Noga Alon and Sepehr Assadi (arXiv). A basic fact from graph theory is that any graph has a \((\Delta+1)\)-coloring, where \(\Delta\) is the maximum degree. Followers of property testing are likely familiar with a fantastic result of Assadi-Chen-Khanna (ACK) on sublinear algorithms, that gives a sublinear algorithm for \((\Delta+1)\)-coloring. (The running time is \(\widetilde{O}(n^{3/2})\), where \(n\) is the number of vertices.) The key tool is a palette sparsification theorem: suppose each vertex is given a “palette” of \((\Delta+1)\) colors. Each vertex randomly sparsifies its palette by sampling \(O(\log n)\) colors, and is constrained to only use these colors. Remarkably, whp the graph can still be properly colored. This tool is at the heart of sublinear time/space algorithms for coloring. This paper gives numerous extensions to this theorem, where one can tradeoff a larger initially palette for a smaller final sample. Another extension is for triangle-free graphs, where the initial palette is of size \(O(\Delta/\ln \Delta)\) and the sample is of size \(O(\Delta^\gamma + \sqrt{\ln n})\) (for parameter \(\gamma < 1\). This leads to an \(O(n^{3/2 + \gamma})\) time algorithm for \(O(\Delta/\ln \Delta)\) coloring of triangle-free graphs.

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time by Sepehr Assadi and Shay Solomon (arXiv). Taking off from sublinear coloring algorithms, one can ask if there are sublinear time algorithms for Maximal Independent Set (MIS) and Maximal Matching (MM). Alas, ACK prove that this is impossible. This paper investigates when one can get a sublinear time algorithm for these problems. For graph \(G\), let \(\beta(G)\) be the “neighborhood independence number”, the size of the largest independent set contained in a vertex neighborhood. This papers shows that both problems can be solved in \(\widetilde{O}(n \beta(G))\) time. Examples of natural classes of graphs where \(\beta(G)\) is constant: line graphs and unit-disk graphs. An interesting aspect is that MIS algorithm is actually deterministic! It’s the simple marking algorithm that rules out neighborhoods of chosen vertices; the analysis shows that not much time is wasted in remarking the same vertex.

Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation by Yu Chen, Sampath Kannan, and Sanjeev Khanna (arXiv). This paper studies sublinear algorithms for the metric TSP problem. The input is an \(n \times n\) distance matrix. One can 2-approximate the TSP by computing the MST, and a result of Czumaj-Sohler gives a \((1+\varepsilon)\)-approximation algorithm for the latter, running in \(O(n\varepsilon^{-O(1)})\) time. The main question is: can one beat the 2-factor approximation in sublinear time? This paper considers the graphic TSP setting, where the distance matrix corresponds to the shortest path metric of an unweighted graph. One result is a \((2-\varepsilon_0)\)-approximation algorithm (for an explicit constant \(\varepsilon_0\)) that runs in \(\widetilde{O}(n)\) time. For the important \((1,2)\) TSP setting (all distances are either 1 or 2), the paper gives a \(O(n^{1.5})\) time 1.63-approximation algorithm. Interestingly, there is a lower bound showing that \((1+\varepsilon)\)-approximations, for arbitrarily small \(\varepsilon\), cannot be achieved in \(o(n^2)\) time. One of the key tools is sublinear algorithms for estimating the maximum matching size, itself a well-studied problem in the community.