Welcome to the posting for first month of 2024! We hope you had a good start to this year. Last month, we had 3 papers. So, without further delay, let’s get started.

On locally-characterized expander graphs (a survey) by Oded Goldreich (ECCC) In this paper, written in an expository style, Goldreich surveys the result of Adler, Kohler and Peng which we covered in our posts on May 2021 and August 2020. The survey begins by reminding the reader what a locally-characterizable graph property is. A family of graphs \(\mathcal{G}\) is said to be characterized by a finite class \(\mathcal{F}\) of graphs if every graph \(G \in \mathcal{G}\) is \(F\)-free for all \(F \in \mathcal{G}\). Thanks to its connections with expressibility in first order logic, one would expect the a locally-characterizable graph property to admit testers depending only on the proximity parameter (in the bounded degree graph model). So, it was quite a surprise when Adler, Kohler and Peng showed that there are locally characterizable graph properties which provably require testers whose query complexity increases with the size of the graph. The main theorem of Adler, Kohler and Peng describes the locally characterizable property that is hard to test. This characterization asserts that you can get your hands on a finite class \(\mathcal{F}\) of graphs so that \(\mathcal{F}\)-freeness of a graph \(G\) means that the graph is a (bounded-degree) expander. One of the key ingredients used in this proof is the Zig-Zag construction of Reingold, Vadhan and Wigderson.

Fast Parallel Sampling under Isoperimetry by Nima Anari, Sinho Chewi, and Thuy-Duong Vuong (arXiv) The featured paper considers the task of sampling (in parallel) from a continuous distribution \(\pi\) supported over \(\mathbb{R}^d\). The main result in the paper shows that for distributions which satisfy a log-Sobolev inequality, you can use parallelized Langevin Algorithms and obtain samples from a distribution \(\pi’\) where \(\pi’\) and \(\pi\) are close in KL-divergence. As an application of their techniques, the authors show that their results can be used to do obtain RNC samples for directed Eulerian Tours and asymmetric Determinantal Point Processes.

Holey graphs: very large Betti numbers are testable by Dániel Szabó, Simon Apers (arXiv) This paper considers the problem of testing whether an input graph \(G\) has a very large \(k\)-th Betti Number (at least \((1-\delta) d_k\) where \(d_k\) denotes the number of \(k\)-cliques in \(G\) and \(\delta > 0\) is sufficiently small) in the dense graph model. As the title indicates, the result says that this property is testable for constant \(k\). Earlier in 2010, Elek investigated this question in the bounded degree model. Elek’s main result showed that with a number of queries \(q = q(\varepsilon\), you can obtain an estimate to the \(k\)-th Betti Number which is within an additive \(\pm \varepsilon n\) of the true \(k\)-th Betti Number. Let us contrast this result with the setting of dense graphs. The authors note that in the dense graph model, getting an estimate to the \(k\)-th Betti Number which is within an additive $\pm \varepsilon d_k$ of the true value needs \(\Omega(n)\) queries. This is the reason why the authors consider the formulation above.

Sorry for delay in getting this month’s post out. This time we had seven (EDIT) eight (LATER EDIT) nine papers. Thanks to our readers for pointing out a paper we missed. Do let us know if we missed any (EDIT) others. Alright, without further delay, let us look at this month’s spread.

Mildly Exponential Lower Bounds on Tolerant Testers for Monotonicity, Unateness, and Juntas by Xi Chen, Anindya De, Yuhao Li, Shivam Nadimpalli, and Rocco A. Servedio (arXiv) Let us begin with a paper on property testing of boolean functions over the binary hypercube \(\{0,1\}^n\). Quoting from the abstract.

[This paper gives] the first superpolynomial (in fact, mildly exponential) lower bounds for tolerant testing of monotonicity, unateness, and juntas with a constant separation between yes and no cases.

Let us take a little dip to get a superficial feel for what techniques the paper uses for boolean monotonicity. The key lies in an adaptation of the lower bound techniques pioneered in the paper of Pallavoor-Raskhodnikova-Waingarten (PRW) which considers distributions over “yes functions” (which are \(\varepsilon/\sqrt{n}\)-close to monone) and “far functions” (which are $\eps$-far from being monotone). PRW split the \(n\) variables into \(n/2\) control variables and \(n/2\) action variables. PRW consider the subcubes indexed by bit-strings where the control variables are balanced. The function value is chosen carefully on these subcubes which ensures any tester that reliably distinguishes yes and no functions need to sample bit-strings from the same balanced subcube which differ in lots of action variables. As PRW argue, this event occurs with low probability if the allowed number of queries is small. The key insight of the featured paper is to modify the PRW construction by using random monotone DNF formulas due to Talagrand. If this sets up your appetite, go read the paper!

Testing Junta Truncation by William He and Shivam Nadimpalli (arXiv)

Let \(f \colon \{0,1\}^n \to \{0,1\}\) be a \(k\)-junta. You are given a set \(\mathcal{U}_{\text{yes}} = \bigl\{0,1\bigr\}^n\) and a set \(\mathcal{U}_{\text{no}} = \bigl\{x \in \{0,1\}^n \colon f(x) = 1 \bigr\}.\) Consider uniform distributions supported on both of these sets which we call \(\mathcal{D}_{\text{yes}}\) and \(\mathcal{D}_{\text{no}}\). Finally, we define a distribution \(\mathcal{D} = \begin{cases} \mathcal{D}_{\text{yes}} \text{ w.p. } 1/2 \\ \mathcal{D}_{\text{no}} \text{ w.p } 1/2 \end{cases}.\)

You are given \(t\) samples from the distribution \(D\). The task is to decide whether \(D\) is the yes distribution or the no distribution. The featured paper shows you can reliably solve this task with \(t \leq \min(2^k + \log{n \choose k }, 2^{k/2} \log^{1/2}{n \choose k})\) samples. The paper also supplements this result with a lower bound of \(t \geq \log{n \choose k}\) samples fewer than which cannot be used to reliably distinguish these two distributions. The results suggest that this “testing junta truncation” problem requires learning the set of relevant variables for the junta.

Longest Common Substring and Longest Palindromic Substring in \(\mathbf{\widetilde{O}(\sqrt n)}\) Time by Domenico Cantone, Simone Faro, Arianna Pavone, and Caterina Viola (arXiv) I paraphrase from the abstract of this paper. You know the longest common substring and longest palindromic substring as classic problems in computer science both of which can be solved in linear time using suffix trees. Recently, quantum algorithms were proposed for both of these problems in the query model both of which issue only \(o(n)\) quantum queries. The featured paper notes that this query model has a shortcoming namely when it comes to real life implementation on actual hardware. The current paper address this shortcoming by presenting \(o(n)\) quantum-query algorithms in the circuit model of computation.

Testing properties of distributions in the streaming model by Sampriti Roy and Yadu Vasudev (arXiv) Alright, now let us consider a different twist on distribution testing. Suppose you have a small memory. You obtain a bunch of samples to solve some standard distribution testing task but the twist is of course you cannot store all the samples. What can you say about how sample complexity trades off against space complexity? The featured paper studies this trade off in the standard access model and the conditional access model. One of the results of the paper asserts that in the conditional access model, you can do identity testing with \( \widetilde{O}\bigl(\frac{\log^4n}{\varepsilon^4}\bigr)\) samples while using only \(O\bigl(\frac{\log^2 n}{\varepsilon^2} \bigr)\) bits of memory.

Testing Spreading Behavior in Networks with Arbitrary Topologies by Augusto Modanese and Yuichi Yoshida (arXiv) We covered the problem of testing dynamic environments in this March 2014 post and that May 2021 post earlier. The goal here is to check whether a dynamically evolving system evolves according to some fixed rule or whether it evolves according to some fixed rule or whether the system is far from systems that evolve according to that fixed rule. The May 2021 post covered a paper which shows you can test dynamically evolving systems that evolve according to what is called the threshold rule. The featured paper considers rules motivated by some kind of models for infection spreading. One of the results in the paper presents one-sided and two-sided testers (with \(O(1/\varepsilon)\) query complexity) for testing a single step of evolution (on bounded degree graphs) with these rules.

A Tight Lower Bound of Ω(log n) for the Estimation of the Number of Defective Items by Nader Bshouty and Gergely Harcos (ECCC) The featured paper considers a problem in group testing. Let us quickly review the setup for group testing. You are given some ground set \(X\) of \(|X| = n\) items. Suppose the set of items in the set \(I \subseteq X\) is defective. The challenge is to devise a test which refers to some set \(Q \subseteq X\) where the test is said to be successful iff \(Q \cap I \neq \emptyset\). This paper presents lower bounds for non-adpative algorithms for group testing. And as the title says, if your algorithm wishes to estimate the number of defective items to within a constant factor, you better pay up \(\Omega(\log n)\) tests.

A tight lower bound on non-adaptive group testing estimation by Tsun-Ming Cheung, Hamed Hatami, and Anthony Ostuni (arXiv) As our readers pointed out, this paper is concurrent with the paper above and achieves the same lower bound. Indeed, this holds for both the one-sided and the two-sided variants. Furthermore, as this paper shows if one knows the set \(I\) satisfies \(L \leq |I| \leq U\) then you can show both one-sided and two-sided lower bounds of \(\Omega(U/L)\) non-adaptive queries if you want a constant approximation to \(|I|\).

On Testing Isomorphism to a Fixed Graph in the Bounded-Degree Graph Model by Oded Goldreich and Laliv Tauber (ECCC) This paper looks like a fantastic read for your students — especially when written in such an engaging style spanning (only) 18 highly readable pages. As the title indicates, this paper considers the challenge of testing isomorphism to a fixed graph in the bounded degree model. The main result of this paper asserts that for almost all \(d\)-regular graphs \(H\) on \(|V(H)| = n\) vertices, testing isomorphism to \(H\) can be done in about \(\approx \sqrt n\) queries. The paper also presents an almost matching (query) lower bound which also holds for almost all graphs \(H\).

Tolerant Testing of High-Dimensional Samplers with Subcube Conditioning by Gunjan Kumar, Kuldeep S. Meel, and Yash Pote (arXiv) Let us consider the distribution testing setup with a twist: Suppose you are given some unknown distribution \(\mathcal{Q}\) supported over \(\{0,1\}^n\) and you want to sample from it conditioned on some predicate \(\mathcal{Q}\). The question is can you efficiently check whether these are legit samples (satisfying the predicate) which are taken from the distribution \(\mathcal{Q}\). Our October 2020 news covered a tolerant tester on this problem which involved some subset of the authors of the current paper. The featured paper considers what additional leverage you gain if you are given access to a sampling oracle which can sample from “conditioned subcubes.” In this model, you can query some subcube and after issuing a query, you will receive an element \(x\) from this subcube with probability proportional to original probability weight of \(x\). The paper provides a tolerant tester in this setup which makes at most \(\widetilde{O}(n^3/(\varepsilon_2 – \varepsilon_1)^5)\) queries to this sampling oracle.

Apologies, dear readers for the delay in getting out this month’s post. This month we had three papers — all on testing properties of graphs! (EDIT: Updated later) four papers: three on property testing problems on graphs and the last one on testing convexity. One of the featured papers this month revisits the problem of testing the properties of directed graphs and comes back with a happy progress report. Alright, let’s dig in.

A Distributed Conductance Tester Without Global Information Collection by Tugkan Batu, Chhaya Trehan (arXiv) One of the classic questions in property testing considers the task of testing expansion. Here, you are interested in knowing whether the input graph has conductance at least \(\alpha\) or it is far from having conductance at most \(\alpha^2\). On a parallel track, we recall that thanks to the classic work of Parnas and Ron, we know there are connections between distributed algorithms and graph property testing. Meditating on these connections led to the emergence of distributed graph property testing as an active area of research. The featured paper considers the task of testing expansion in the distributed framework. The algorithms presented give a distributed implementation of multiple random walks from all vertices, and controls the congestion of the implementation. In particular, this leads to a \(O(\log n/\alpha^2)\) round expansion-tester. In a first attempt at such an implementation, you might note that you need to track how well short random walks mix when started from a bunch of randomly chosen vertices. This seems to require maintaining some global state/global aggregate information. One of the important features of their algorithm (as mentioned in the title) does away with the need to maintain such global states. As a closing remark, I note the algorithm presented in this paper does not require the graph to be bounded degree.

Testing versus estimation of graph properties, revisited by Lior Gishboliner, Nick Kushnir, Asaf Shapira (arXiv) This paper considers the task of additively estimating the distance to a property \(\mathcal{P}\) of a dense graph. Let me set up some context to view the results in the featured paper by summarizing what was known before. One of the early important results in this area is the original result of Fischer and Newman which shows that any testable graph property admits a \(\pm \varepsilon\) distance approximation algorithm, which follows from the regularity lemma. However, the query complexity of the resulting algorithm is a Wowzer-style bound. Later, Hoppen et al., building upon tools from the classic work of Conlon and Fox, demonstrated that this bound of \(twr(poly(1/\varepsilon))\) also holds for testable hereditary properties. Fiat and Ron introduced a decomposition machinery that allowed them to decompose a “complex” property into a collection of simpler properties. They used this decomposition to estimate distances to a vast collection of graph properties. They also asked if it was possible to find a transformation using which one can bypass the blowup in query complexity incurred by Fischer and Newman for some rich class of graph properties. The featured paper proves that for a hereditary graph property, you can in fact get algorithms where the query complexity for distance estimation only grows as \(\exp(1/\varepsilon)\). Additionally, for every testable graph property, you can get distance estimators for that property whose query complexity only grows doubly exponentially in \(1/\varepsilon\) (as opposed to the tower bound earlier).

An Optimal Separation between Two Property Testing Models for Bounded Degree Directed Graphs by Pan Peng, Yuyang Wang (arXiv) Unlike undirected graphs, directed graph properties have not received as much attention in the property testing community. In a classic work, Bender and Ron considered two natural models for studying property testing on directed graphs. The first model is one where you can only follow the “out” edges or the so-called unidirectional model. In the other model, you are allowed to follow both the “out” edges and the “in” edges incident on the vertex which is also called the bidirectional model. The featured paper considers directed graphs where the in-degree and the out-degrees are both bounded in both of the models mentioned above. The graph is presented to you in the adjacency list format (tailored for the model you consider). The paper shows that even for the fundamental task of subgraph-freeness, the directed world has some interesting behavior with respect to the two models above. Let me showcase one of the catchy results from the paper which illustrates this separation nicely. Take a connected directed graph \(H\) with \(k\) source components. The paper shows that for sufficiently small \(\varepsilon\), testing whether a directed graph \(G\) is \(H\)-free or \(\varepsilon\)-far from being \(H\)-free requires \(\Omega(n^{1-1/k})\) unidirectional queries. On the other hand, in the bidirectional model, this can be done with a mere \(O_{\varepsilon, d, k}(1)\) number of queries.

(EDIT: Added later)

Testing Convexity of Discrete Sets in High Dimensions by Hadley Black, Eric Blais, Nathaniel Harms (arXiv) As the title suggests, this paper explores the problem of testing convexity. To understand the notion of convexity explored in the paper, consider the following setup: You call a set \(S \subseteq \{-1, 0, 1\}^n\) convex if there exists a convex set \(S’ \subseteq \mathbb{R}^n\) such that \(S = S’ \cap \{-1,0,1\}^n\). And you call a set \(S \subseteq \{-1,0,1\}^n\) far from being convex if for every convex \(T \subseteq \{-1,0,1\}^n\), you have \(|S \oplus T| \geq \varepsilon 3^n\). The paper shows that when you are allowed membership queries, you can test convexity non-adaptively with a one-sided error by issuing \(3^{O(\sqrt{n \log(1/\varepsilon)})}\) queries. Also, they prove an almost matching lower bound. Finally, with a lower bound of \(3^{\Omega(n)}\) when confined to using sample-based testers, authors provably demonstrate that membership queries indeed buy you some undeniable power for testing convexity in high dimensions.

After an empty month, the engines of PTReview are roaring back to life with a fresh batch of papers for this month’s edition. In total, we have four papers that are sure to pique your interest. It’s been an action-packed month with a diverse range of topics covered in the featured papers. The first paper delves into new variations in distribution testing, while the second paper discusses optimal testers for Bayes Nets. The third paper focuses on optimal tolerant junta-testers, and the final paper presents a cool monotonicity tester over hypergrids.

Distribution Testing Under the Parity Trace by Renato Ferreira Pinto Jr and Nathaniel Harms (arXiv) The featured paper considers the classic setup in distribution testing with a twist. To explain the results, let me introduce the framework considered in this work. Consider distributions supported over \([n]\). Suppose I want to design distribution testers where instead of obtaining samples in the usual way, I first obtain an ordered list of samples, I store it in a sequence \(S\) and only the least significant bit of each element of \(S\) is made available to your distribution testing algorithm. This is called a parity trace. Note that with this access model, suddenly a bunch of standard tasks become non-trivial. To take an example from the paper, you can no longer distinguish between the uniform distribution supported on \(\{1,2, \ldots, n\}\) and the uniform distribution supported on \(\{n+1, n+2, \ldots 2n\}\) in this access model. Nevertheless, the paper shows, you can still obtain testers which require only sublinear number of accesses for testing uniformity in this model.

Another cool feature of this big paper is an unexpected foray into the trace reconstruction literature from a property testing viewpoint. I wish I understood the formal connection better to describe a bit more about it. But for now, let me leave that as an appetizer which (hopefully) encourages you to take a look at the paper.

New Lower Bounds for Adaptive Tolerant Junta Testing by Xi Chen and Shyamal Patel (arXiv) If you are a regular here on the PTReview corner, you are probably no stranger to the tolerant junta testing problem. As a corollary to the main result, the paper in question proves a lower bound of \(k^{\Omega(\log k)}\) queries on any adaptive algorithm that wishes to test whether the input function \(f\) is \(\varepsilon_1\) close to being a \(k\)-junta or whether it is \(\varepsilon_2\)-far \(\left(\text{where } \varepsilon_2 \geq \varepsilon_1 + \displaystyle\frac{1}{poly(k)}\right)\). Indeed, another remarkable achievement of the paper is that it achieves a superpolynomial separation between non-tolerant versions and the tolerant versions of any natural property of boolean functions under the adaptive setting.

Near-Optimal Degree Testing for Bayes Nets by Vipul Arora, Arnab Bhattacharyya, Clément L. Canonne (our own!) and Joy Qiping Yang (arXiv) This paper continues a line of investigation which a subset of the authors were a part of (which we also covered in our News for April 2022). Let us remind ourselves of the setup. You are given a probability distribution \(\mathcal{P}\) supported over the Boolean Hypercube. Suppose \(\mathcal{P}\) can be generated by a Bayseian Network. You may think of a Bayesian Network as a DAG where each vertex tosses a coin (with different heads probabilities). The question seeks to test whether \(\mathcal{P}\) admits a sparse Bayesian Net (in the sense of each vertex having small in-degree). The main result of the paper gives an algorithm for this task which requires \(\Theta(2^{n/2}/\varepsilon^2)\) samples. The paper also proves an almost matching lower bound.

A \(d^{1/2+o(1)}\) Monotonicity Tester for Boolean Functions on \(d\)-Dimensional Hypergrids by Hadley Black, Deeparnab Chakrabarty and C. Seshadhri (again, our own!) (arXiv) Again, the problem of monotonicity testing of boolean functions hardly requires any detailing to the regular readers of PTReview. As you can see in our News from November 2022 there were two concurrent papers mulling over this problem over the hypergrid domain. The featured paper is the result of a dedicated pursuit by the authors and the key result is what the title says. Namely, you can test monotonicity with a number of (non-adaptive, one-sided) queries that has no dependence on \(n\).

All the best to everyone for a Happy 2023. The holiday season is ripe with lots of papers to engage our readers. So, we have nine papers and we hope some of those will catch your fancy. As a new year treat, we also feature Gilmer’s constant lower bound on the union-closed sets problem of Frankl. On we go.

Sublinear Time Algorithms and Complexity of Approximate Maximum Matching by Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein (arXiv) This paper makes significantly advances our understanding of the maximum matching problem in the sublinear regime. Your goal is to estimate the size of the maximum matching and you may assume that you have query access to the adjacency list of your graph. Our posts from Dec 2021 and June 2022 reported some impressive progress on this problem. The upshot from these works essentially said that you can beat greedy matching and obtain a \(\frac{1}{2} + \Omega(1)\) approximate maximum matching in sublinear time. Let me first go over the algorithmic results from the current paper. The paper shows the following two algorithmic results:

(1) An algorithm that runs in time \(n^{2 – \Omega_{\varepsilon}(1)}\) and returns a \(2/3 – \varepsilon\) approximation to maximum matching in general graphs, and

(2) An algorithm that runs in time \(n^{2 – \Omega_{\varepsilon}(1)}\) and returns a \(2/3 + \varepsilon\) approximation to maximum matching size in bipartite graphs.

The question remained — can we show a lower bound that grows superlinearly with \(n\). The current work achieves this and shows that even on bipartite graphs, you must make at least \(n^{1.2 – o(1)}\) queries to the adjacency list to get a better than \(2/3 + \Omega(1)\) approximation. (An aside: A concurrent work by Bhattacharya-Kiss-Saranurak from December also obtains similar algorithmic results for approximating the maximum matching size in general graphs).

Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an \(\widetilde{O}(n \sqrt d)\) Monotonicity Tester by Hadley Black, Deeparnab Chakrabarty, C. Seshadhri (arXiv) Boolean Monotonicity testing is as classic as classic gets in property testing. Encouraged by the success of isoperimetric theorems over the hypercube domain and the monotonicity testers powered by these isoperimetries (over the hypercube), one may wish to obtain efficient monotonicity testers for the hypergrid \([n]^d\). Indeed, the same gang of authors as above showed in a previous work that a Margulis style directed isoperimetry can be extended from the lowly hypercube to the hypergrid. This resulted in a tester with \(\widetilde{O}(d^{5/6})\) queries. The more intricate task of proving a directed Talagrand style isoperimetry that underlies the Khot-Minzer-Safra breakthrough was a challenge. Was. The featured work extends this isoperimetry from the hypercube to the hypergrid and this gives a tester with query complexity \(\widetilde{O}(n \sqrt d)\) which is an improvement over the \(d^{5/6}\) bound for domains where \(n\) is (say) some small constant. But as they say, when it rains, it pours. This brings us to a concurrent paper with the same result.

Improved Monotonicity Testers via Hypercube Embeddings by Mark Braverman, Subhash Khot, Guy Kindler, Dor Minzer (arXiv) Similar to the paper above, this paper also obtains monotonicity testers over the hypergrid domain, \([n]^d\), with \(\widetilde{O}(n^3 \sqrt d)\) queries. This paper also presents monotonicity testers over the standard hypercube domain — \(\{0,1\}^d\) in the \(p\)-biased setting. In particular, their tester issues \(\widetilde{O}(\sqrt d)\) queries to successfully test monotonicity on the \(p\)-biased cube. Coolly enough, this paper also proves directed Talagrand style isoperimetric inequalities both over the hypergrid and the \(p\)-biased hypercube domains.

Toeplitz Low-Rank Approximation with Sublinear Query Complexity by Michael Kapralov, Hannah Lawrence, Mikhail Makarov, Cameron Musco, Kshiteej Sheth (arXiv) Another intriguing paper for the holiday month. So, take a Toeplitz matrix. Did you know that any psd Toeplitz matrix admits a (near-optimal in the Frobenius norm) low-rank approximation which is itself Toeplitz? This is a remarkable statement. The featured paper proves this result and uses it to get more algorithmic mileage. In particular, suppose you are given a \(d \times d\) Toeplitz matrix \(T\). Armed with the techniques from the paper you get algorithms that return a Toeplitz matrix \(\widetilde{T}\) with rank slightly bigger than \(rank(T)\) which is a very good approximation to \(T\) in the Frobenius norm. Moreover, the algorithm only issues a number of queries sublinear in the size of \(T\).

Sampling an Edge in Sublinear Time Exactly and Optimally by Talya Eden, Shyam Narayanan and Jakub Tětek (arXiv) Regular readers of PTReview are no strangers to the fundamental task of sampling a random edge from a graph which you can access via query access to its vertices. Of course, you don’t have direct access to the edges of this graph. This paper considers the task of sampling a truly uniform edge from the graph \(G = (V,E)\) with \(|V| = n, |E| = m\). In STOC 22, Tětek and Thorup presented an algorithm for a relaxation of this problem where you want an \(\varepsilon\)-approximately unifrom edge. This algorithm runs in time \(O\left(\frac{n}{\sqrt{m}} \cdot \log(1/\varepsilon) \right)\). The featured paper presents an algorithm that samples an honest to goodness uniform edge in expected time \(O(n/\sqrt{m})\). This closes the problem as we already know a matching lower bound. Indeed, just consider a graph with \(O(\sqrt m)\) vertices which induce a clique and all the remaining components are singletons. You need to sample at least \(\Omega(n/\sqrt m)\) vertices before you see any edge.

Support Size Estimation: The Power of Conditioning by Diptarka Chakraborty, Gunjan Kumar, Kuldeep S. Meel (arXiv) This work considers the classic problem of support size estimation with a slight twist. You are given access to a stronger (conditioning based) sampling oracle. Let me highlight one of the results from this paper. So, you are given a distribution \(D\) where \(supp(D) \subseteq [n]\). You want to obtain an estimate to \(supp(D)\) that lies within \(supp(D) \pm \varepsilon n\) with high probability. Suppose you are also given access to the following sampling oracle. You may choose any subset \(S \subseteq [n]\) and you may request a sample \(x \sim D\vert_S\). An element \(x \in S\) is returned with probability \(D\vert_S(x) = D(x)/D(S)\) (for simplicity of this post, let us assume \(D(S) > 0\)). In addition, this oracle also reveals for you the value \(D(x)\). The paper shows that the algorithmic task of obtaining a high probability estimate to the support size (to within \(\pm \varepsilon n\)) with this sampling oracle admits a lower bound of \(\Omega(\log (\log n)\) calls to the sampling oracle.

Computing (1+epsilon)-Approximate Degeneracy in Sublinear Time by Valerie King, Alex Thomo, Quinton Yong (arXiv) Degeneracy is one of the important graph parameters which is relevant to several problems in algorithmic graph theory. A graph \(G = (V,E)\) is \(\delta\)-degenerate if all induced subgraphs of \(G\) contain a vertex with degree at most \(\delta\). The featured paper presents algorithms for a \((1 + \varepsilon)\)-approximation to degeneracy of \(G\) where you are given access to \(G\) via its adjacency list.

Learning and Testing Latent-Tree Ising Models Efficiently by Davin Choo, Yuval Dagan, Constantinos Daskalakis, Anthimos Vardis Kandiros (arXiv) Ising models are emerging as a rich and fertile frontier for Property Testing and Learning Theory researchers (at least to the uninitiated ones like me). This paper considers latent-tree ising models. These are ising models that can only be observed at their leaf nodes. One of the results in this paper gives an algorithm for testing whether the leaf distributions attached to two latent-tree ising models are close or far in the TV distance.

A constant lower bound for the union-closed sets conjecture by Justin Gilmer (arXiv) The union-closed sets conjecture of Frankl states that for any union closed set system \(\mathcal{F} \subseteq 2^{[n]}\), it holds that there is a mysterious element \(i \in [n]\) that shows up in at least \(c = 1/2\) of the sets in \(\mathcal{F}\). Gilmer took a first swipe on this problem and gave a constant lower bound of \(c = 0.01\). This has already been improved by at least four different groups to \(\frac{3-\sqrt{5}}{2}\), a bound which is the limit of Gilmer’s method (which takes all of only 9 pages!).

The key lemma Gilmer proves is the following. Suppose you sample two sets: \(A, B \sim \mathcal{D}_n\) (iid) from some distribution \(\mathcal{D}_n\) over the subsets of \([n]\). Suppose for every index \(i \in [n]\), it holds that the probability that the element \(i\) shows up in the random set \(A\) is at most $0.01$. Then you have \(H(A \cup B) \geq 1.26 H(A)\). This is all you need to finish Gilmer’s proof (of \(c = 0.01\)). The remaining argument is as follows. Suppose, by the way of contradiction, that no element shows up in at least \(0.01\) fraction of sets in the union closed family \(\mathcal{F}\). An application of the key lemma would then give \(H(A \cup B) > H(A)\) which is a contradiction if \(A,B\) are chosen uniformly from \(\mathcal{F}\). The proof of the key lemma is also fairly slick and uses pretty simple information theoretic tools.

Last month saw a flurry of activity in Property Testing. We had thirteen papers!! Without further ado, let us dig in.

Testing of Index-Invariant Properties in the Huge Object Model (by Sourav Chakraborty, Eldar Fischer, Arijit Ghosh, Gopinath Mishra, and Sayantan Sen)(arXiv) This paper explores a class of distribution testing problems in the Huge Object Model introduced by Goldreich and Ron (see our coverage of the model here). A quick refresher of this model: so, suppose you want to test whether a distribution \(\mathcal{D}\) supported over, say the boolean hypercube \(\{0,1\}^n\) has a certain property \(\mathcal{P}\). You pick a string \(x \sim \mathcal{D}\) where the length of \(x\) is \(n\). In situations where \(n\) is really large, you might not want to read all of \(x\) and you may instead want to read only a few bits from it. To this end, Goldreich and Ron formulated a model where you have query access to the strings you sample. The distribution \(\mathcal{D}\) is deemed to be \(\varepsilon\)-far from \(\mathcal{P}\) if \(EMD(\mathcal{D}, \mathcal{P}) \geq \varepsilon\) (here \(EMD\) denotes the earthmover distance with respect to the relative Hamming distance between bitstrings). In this model, one parameter of interest is the query complexity of your tester.

One of the results in the featured paper above shows the following: Let \(\sf{MONOTONE}\) denote the class of monotone distributions supported over \(\{0,1\}^n\) (a distribution \(D\) belongs to the class \(\sf{MONOTONE}\) if \(D(x) \leq D(y)\) whenever \(0^n \preceq x \preceq y \preceq 1^n\)). Let \(\mathcal{B}_d\) denote the class of distributions supported over \(\{0,1\}^n\) whose supports have VC dimension at most \(d\). Let \(\mathcal{P} = \sf{MONOTONE} \cap \mathcal{B}_d\). Then, for any \(\varepsilon > 0\), you can test whether a distribution \(\mathcal{D} \in \mathcal{P}\) or whether it is \(\varepsilon\) far from \(\mathcal{P}\) with query complexity \(poly(1/\varepsilon)\). In fact, the paper shows this for a much richer class \(\mathcal{P}\) which is the class of so-called index-invariant distributions with bounded VC-dimensions. The paper also shows the necessity of both of these conditions for efficient testability. Do check it out!

Identity Testing for High-Dimensional Distributions via Entropy Tensorization (by Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda)(arXiv)

This paper considers a classic in distribution testing. Namely, the problem of testing whether the hidden input distribution \(\pi\) is identical to an explicitly given distribution \(\mu\). Both distributions are supported over a set \(\Omega\). The caveat is \(\Omega\) is some high dimensional set (think \(\Omega = [k]^n\)) and that it has a size that grows exponentially in \(n\). In this case, identity testing has sample complexity \(\Omega(k^{n/2})\) even when \(\mu\) is the uniform distribution. In an attempt to overcome this apparent intractability of identity testing in high dimensions, this paper takes the following route: in addition to the standard sample access to \(\pi\), you also assume access to a stronger sampling oracle from \(\pi\). And now you would like to understand for which class of explicitly given distributions \(\mu\) can you expect algorithms with efficient sample complexity (assuming the algorithm is equipped with this stronger sampling oracle). For any \(i \in [n]\) and \(\omega \in \Omega\), the stronger oracle considered in this work allows you to sample \(x \sim \pi_{\omega(-i)}\) where \(\pi_{\omega(-i)}\) denotes the conditional marginal distribution of \(\pi\) over the \(i\)-th coordinate when the remaining coordinates have been fixed according to \(\omega\).

The paper shows if the known distribution \(\mu\) satisfies some approximate tensorization of entropy criterion, then identity testing with such distributions \(\mu\) can be done with \(\tilde{O}(n/\varepsilon)\) queries. Thanks to the spectral independence toolkit pioneered by Anari et al, it turns out that the approximate tensorization property holds for a rich class of distributions. (A side note to self: It looks like I am running out of reasons to postpone learning about the new tools like Spectral Independence.)

Near-Optimal Bounds for Testing Histogram Distributions (by Clément L. Canonne, Ilias Diakonikolas, Daniel M. Kane, and Sihan Liu)(arXiv) Histograms comprise one of the most natural and widely used ways for summarizing some relevant aspects of massive datasets. Let \(\Omega\) denote an \(n\)-element dataset (with elements being \(\{1,2, \ldots, n \}\)). A \(k\)-histogram is a function that is piecewise constant over \(k\) interval pieces. This paper studies the sample complexity of the following fundamental task: given a distribution \(\mathcal{P}\) supported over \(\Omega\), is \(\mathcal{P}\) a \(k\)-histogram or is \(\mathcal{P}\) far from being a \(k\)-histogram. The main result of the paper is a (near) sample optimal algorithm for this problem. Specifically, this paper shows that \(k\)-histogram testing has sample complexity \(\Theta\left(\sqrt{nk}/\varepsilon + k/\varepsilon^2 + \sqrt{n}/\varepsilon^2\right)\).

Comments on “Testing Conditional Independence of Discrete Distributions” (by Ilmun Kim)(arXiv) Probability is full of subtleties and conditional probability is perhaps the biggest landmine of subtleties in this venerable discipline. The featured paper closely examines some subtleties in Theorem 1.3 of the CDKS18 paper on testing conditional independence of discrete distributions. Essentially, this theorem undertakes the following endeavor: you would like to test whether a bivariate discrete distribution has independent marginals conditioned on values assumed by a third random variable. Theorem 1.3 of CDKS18 asserts that there exists a computationally efficient tester for conditional independence with small sample complexity. The featured paper fixes the sample complexity bound claimed in Theorem 1.3 of CDKS18.

Cryptographic Hardness of Learning Halfspaces with Massart Noise (by Ilias Diakonikolas, Daniel M. Kane, Pasin Manurangsi, and Lisheng Ren)(arXiv) The study of robust supervised learning in high dimensions has seen a lot of impressive progress in the last few years. The paper under review presents sample complexity lower bounds for the task of learning halfspaces in this overarching framework. Let us unpack this paper slowly. So, let us recall the classic task of learning halfspaces in \(\mathbb{R}^n\). You know the drill. I have a known concept class \(\mathcal{C}\) (comprising of boolean functions) in my hand. Unbeknownst to you, I have a boolean function \(f \in \mathcal{C}\). You get as input a multiset \(\{x_i, f(x_i)\}_{i \in [s]}\) of labeled examples from a distribution \(\mathcal{D}\) where \(x_i \sim \mathcal{D}_x\) and \(\mathcal{D}_x\) is fixed but arbitrary. Your goal is to develop an algorithm that returns a hypothesis with a small misclassification rate. The classic stuff.

Now, consider the same setup with a little twist: the so-called Massart noise setup. The labels \(f(x_i)\) are no longer reliable and the label on each \(x_i\) gets flipped adversarially with probability \(\eta_i \leq \eta < 1/2\). In a breakthrough Diakonikolas, Gouleakis, and Tzamos made the first algorithmic progress on this problem and gave algorithms with running time \(poly(n/\varepsilon)\) and misclassification rate \(\eta + \varepsilon\). The current paper shows a lower-bound result. Assuming the hardness of the so-called “Learning With Errors” problem, this paper shows that under Massart Noise, it is not possible for a polynomial time learning algorithm to achieve a misclassification rate of \(o(\eta)\).

Locally-iterative (Δ+1)-Coloring in Sublinear (in Δ) Rounds (by Xinyu Fu, Yitong Yin, and Chaodong Zheng)(arXiv) A time-honored problem in Distributed Computing is Distributed graph coloring. Let us first understand what problem this paper studies. So, you are given a graph \(G = (V,E)\) with maximum degree \(\Delta\). In a seminal work, Szegedy and Vishwanathan introduced the framework of locally-iterative algorithms as a natural family of distributed graph coloring algorithms. These algorithms proceed in \(r\) rounds. In each round, you update the color of a vertex \(v\) where the new color of \(v\) is a function of the current color of \(v\) and the current color of its neighbors. The current paper shows that you can in the locally-iterative framework, you can in fact, obtain a proper coloring of \(G\) with \(\Delta(G) + 1\) colors in \(r = O(\Delta^{3/4} \log \Delta) + \log^* n\) rounds.

Learning Hierarchical Structure of Clusterable Graphs (by Michael Kapralov, Akash Kumar, Silvio Lattanzi, Aida Mousavifar)(arXiv) [Disclaimer: I am one of the authors of this paper.] Hierarchical clustering of graph data is a fundamentally important task in the current big data era. In 2016, Dasgupta introduced the notion of Dasgupta cost which essentially allows one to measure the quality of a hierarchical clustering. This paper presents algorithms that can estimate the Dasgupta Cost of a graph coming from a special family of \(k\)-clusterable graphs in the semi-supervised setting. These graphs have \(k\) clusters. These clusters are essentially subsets of vertices that induce expanders and these clusters are sparsely connected to each other. We are given query access to the adjacency list of \(G\). Also, for an initial “warmup” set of randomly chosen vertices, we are told the clusters they belong to. Armed with this setup, this paper presents algorithms that run in time \(\approx \sqrt{n}\) and return an estimate to the Dasgupta Cost of \(G\) which is within a \(\approx \sqrt{\log k}\) factor of the optimum cost.

Finding a Hidden Edge (by Ron Kupfer and Noam Nisan)(arXiv) Let us consider as a warmup (as done in the paper) the following toy problem. You have a graph on \(n\) vertices whose edge set \(E\) is hidden from you. Your objective is to return any \((i,j) \in E\). The only queries you are allowed are of the following form. You may consider any subset \(Q \subseteq V \times V\) and you can ask whether \(Q\) contains any edge. A simple binary search solves this question with \(\log m\) queries (where \(m = {n \choose 2}\)). However, if you want a non-adaptive algorithm for this problem (unlike binary search) you can show that any deterministic algorithm must issue \(m\) non-adaptive queries. Turns out randomness can help you get away with only \(O(\log^2m)\) non-adaptive queries for this special toy problem. Now, let me describe the problem considered in this work in earnest. Suppose the only queries you are allowed are of the following form: you may pick any \(S \subseteq V\) and you may ask whether the graph induced on \(S\) contains an edge. The paper’s main result is that there is an algorithm for finding an edge in \(G\) which issues nearly linear in \(n\) many non-adaptive queries. The paper also presents an almost matching lower bound.

On One-Sided Testing Affine Subspaces (by Nader Bshouty)(ECCC) Dictatorship testing is one of the classics in property testing of boolean functions. A more generalized problem considers testing whether the presented function is a \(k\)-monomial. If you are a regular reader of the posts on PTReview, you might have seen this problem essentially asks you to test whether a boolean function \(f \colon \mathcal{F}^n \to {0,1}\) is an indicator of an \((n-d)\) dimensional affine/linear subspace of \(\mathcal{F}^n\) (here \(\mathcal{F}\) denotes a finite field). Namely, you would like to test whether the set \(f^{-1}\) is an \((n-k)\) dimensional affine subspace of \(\mathcal{F}^n\). The paper under review improves the state-of-the-art query complexity for this problem from a previous value of \(O\left(|\mathcal{F}|/\varepsilon\right)\) to \(\tilde{O}\left(1/\varepsilon\right)\).

Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries (by Raghavendra Addanki, Andrew McGregor, and Cameron Musco)(arXiv) If you have been around the PTReview corner for a while, you know that sublinear time estimation of graph properties is one of our favorite pastimes here. Classic work in this area considers the following queries: vertex degree queries, \(i\)-th neighbor queries, and edge existence queries. This classic query model has received a lot of attention and thanks to the work of Eden and Rosenbaum we know algorithms for near-uniform edge sampling with query complexity \(O(n/\sqrt{m}) \cdot poly(\log n) \cdot poly(1/\varepsilon)\). Motivated by a desire to obtain more query-efficient algorithms, Beame et al. introduced an augmented query model where you are also allowed the following queries: you may pick \(L, R \subseteq V\) and you get a yes/no response indicating whether there exists an edge in \(E(L, R)\). These are also called the bipartite independent set (BIS) queries. The featured paper shows that with (BIS) queries you get non-adaptive algorithms for near-uniform edge sampling with query complexity being a mere \(\widetilde{O}(\varepsilon^{-4} \log^6 n)\). The main result of the paper gives a non-adaptive algorithm for estimating the number of edges in \(G\) with query complexity (under BIS) being a mere \(\widetilde{O}(\varepsilon^{-5} \log^5 n)\).

A Query-Optimal Algorithm for Finding Counterfactuals (by Guy Blanc, Caleb Koch, Jane Lange, Li-Yang Tan)(arXiv) Given an abstract space \(X^d\), an instance \(x^* \in X^d\) and a model \(f\) (which you think of as a boolean function over \(X^d\)), a point \(x’ \in X^d\) is called a counterfactual to \(x^*\) if \(x^*, x’\) differ in few features (i.e., have a small Hamming distance) and \(f(x^*) \neq f(x’)\). Ideally, you would like to find counterfactuals that are as close to each other in Hamming Distance. The main result of this paper is the following: Take a monotone model \(f \colon \{0,1\}^d \to \{0,1\}\), an instance \(x^* \in \{0,1\}^d\) with small sensitivity (say \(\alpha\)). Then there exists an algorithm that makes at most \(\alpha^{\Delta(x^*)}\) queries to \(f\) and returns all optimal counterfactuals of \(f\). Here \(\Delta(x^*) = \min_{x \in \{0,1\}^d} \{\Delta_H(x, x^*) \colon f(x) \neq f(x^*) \}\). The paper also proves a matching lower bound on query complexity which is obtained by some monotone model \(f\).

A Sublinear-Time Quantum Algorithm for Approximating Partition Functions (by Arjan Cornelissen and Yassine Hamoudi)(arXiv) For the classical Hamiltonian \(H \colon \Omega \to \{0,1, \ldots, n\}\), at inverse temperature \(\beta\), the probability, under the so-called Gibbs distribution, assigned to a state \(x \in \Omega\) is proportional to \(\exp(-\beta H(x))\). The partition function is given by \(Z(\beta) = \sum_{x \in \Omega} \exp(-\beta H(x))\). At high temperatures (or low values of \(\beta\)) the partition function is typically easy to compute. However, the low-temperature regime is often challenging. You use MCMC methods to compute \(Z(\infty)\). In particular, you write this as the following telescoping product \(Z(\infty) = Z(0) \cdot \prod_{i = 0}^{i = \ell – 1} \frac{Z(\beta_{i+1})}{Z(\beta_i)}\) where \(0 = \beta_1 < \beta_2 < \ldots < \beta_{\ell} = \infty\) is some increasing sequence of inverse temperatures with limited fluctuations in Gibbs distribution between two consecutive values and you use MCMC methods to estimate each of the \(\ell\) ratios in the above product. The main result of this paper presents a quantum algorithm that on input a Gibbs distribution generated by a Markov Chain with a large spectral gap performs sublinearly few steps (in size of the logarithm of the state space) of the quantum walk operator and returns a \(\pm \varepsilon Z(\infty)\) additive estimate to \(Z(\infty)\).

A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation (by Omar Alrabiah, Venkatesan Guruswami, Pravesh Kothari, and Peter Manohar)(ECCC) If you made it till here, it is time for a treat. Let us close (hopefully, I did not miss any papers this time!) with a breakthrough in Locally Decodable Codes. So, for 2-query LDCs, we know fairly tight bounds on the block length. For 3-query LDCs, on the other hand, we know a sub-exponential upper bound on the block length. However, the best-known lower bound on the block length was merely quadratic. The featured paper improves this to a cubic lower bound on the block length. The main tool used to achieve this is a surprising connection between the existence of locally decodable codes and the refutation of Boolean CSP instances with limited randomness. This looks like a fantastic read to close off this month’s report!

This was a relatively sleepy month with only two property testing papers. Do let us know if we missed any. Let us dig in. (EDIT: Two updates.)

1. I missed two papers. One on the estimation of quantum entropies and the other on algorithms and lower bounds for estimating MST and TSP costs.
2. Finally, I forgot to welcome our new editor. Welcome onboard, Nithin Varma!!

Private High-Dimensional Hypothesis Testing by Shyam Narayanan (arXiv) This paper continues the novel study of distribution testing under the constraints brought forth by differential privacy extending the work of Canonne-Kamath-McMillan-Ullman-Zakynthinou (henceforth CKMUZ, covered in our May 2019 post). In particular, the paper presents algorithms with optimal sample complexity for private identity testing of \(d\)-dimensional Gaussians. In more detail, the paper shows that can be done with a mere \(\widetilde{O}\left( \frac{d^{1/2}}{\alpha^2} + \frac{ d^{1/3} }{ \alpha^{4/3} \cdot \varepsilon^{2/3}} + \frac{1}{\alpha \cdot \varepsilon} \right)\). Here \(\alpha\) is the proximity parameter and \(\varepsilon\) is the privacy parameter. Combined with a previous result of Acharya-Sun-Zhang, the paper proves that private identity testing of \(d\)-dimensional Gaussians is doable with a sample complexity smaller than that of private identity testing of discrete distributions over a domain of size \(d\) thereby refuting a conjecture of CKMUZ.

Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds by Badih Ghazi, Ravi Kumar, Pasin Manurangsi and Jelani Nelson (arXiv) Adam Sealfon considered the classic All Pairs Shortest Path Problem (the APSP problem) with privacy considerations in 2016. In the \((\varepsilon, \delta)\)-DP framework, Sealfon presented an algorithm which on input an edge-weighted graph \(G=(V,E,w)\) adds Laplace noise to all edge weights and computes the shortest paths on this noisy graph. The output of the algorithm satisfies that the estimated distance between every pair is within an additive \(\pm O(n \log n/\varepsilon)\) of the actual distance (the absolute value of this parameter is called the accuracy of the algorithm). Moreover, this error is tight up to a logarithmic factor if the algorithm is required to release the shortest paths. The current paper shows you can privately release all the pairwise distances while suffering only a sublinear accuracy if you additionally release the edge weights (in place of releasing the shortest paths). In particular, this paper presents an \(\varepsilon\)-DP algorithm with sublinear \(\widetilde{O}(n^{2/3})\) accuracy.

Quantum algorithms for estimating quantum entropies by Youle Wang, Benchi Zhao, Xin Wang (arXiv) So, remember our post from December on sublinear quantum algorithms for estimation of quantum (von Neumann) entropy? The current paper begins by noting that the research so far (along the lines of the work above) assumes access to a quantum query model for the input state which we do not yet know how to construct efficiently. This paper addresses this issue and gives quantum algorithms to estimate the von Neumann entropy of a \(n\)-qubit quantum state \(\rho\) by using independent copies of the input state.

Sublinear Algorithms and Lower Bounds for Estimating MST and TSP Cost in General Metrics by Yu Chen, Sanjeev Khanna, Zihan Tan (arXiv) As mentioned in the title, this paper studies sublinear algorithms for the metric MST and the metric TSP problem. The paper obtains a wide assortment of results and shows that both these problems admit an \(\alpha\)-approximation algorithm which uses \(O(n/\alpha)\) space. This algorithm assumes that the input is given as a stream of \(n \choose 2\) metric entries. Under this model, the paper also presents an \(\Omega(n/\alpha^2)\) space lower bound. Let me highlight one more result from the paper. In a previous news (from June 2020), we covered a result detailing a better than \(2\)-approximation for the graphic TSP and \((1,2)\) TSP which runs in sublinear time. This paper extends this result and obtains better than \(2\)-approximation for TSP on a relaively richer class of metrics.

The holiday season is just around the corner. Best wishes to you and your loved ones in advance from us here at PTReview. This month, we had four five papers in all. To prepare for the festive mood associated with the onset of December, also included are brief updates on the recently disproven implicit graph conjecture. Let’s dig in. (And please, do let us know if we missed your paper).

(EDIT: Thanks to our readers for pointing out a paper we missed.)

Downsampling for Testing and Learning in Product Distributions by Nathaniel Harms, Yuichi Yoshida (arXiv). Contemplating on connections in algorithms used for testing a diverse collection of function properties, this paper provides a unified and generalized view of a technique: which the authors call downsampling. As the paper explains, the name is motivated by analogy to image/signal processing tasks. Very often, these tasks involve two steps. In the first step, you break the input domain into “grid cells”. You use oracle calls to the function to obtain a crude approximation over all these cells. In the second step, you learn this gridded-up or coarsened function cell-by-cell.

This two-step attack could just be what the doctor ordered for your favorite function property testing problem: in particular, it has been put to work for approximately testing visual properties, approximating distance to monotonicity in high dimensions, testing \(k\)-monotone functions, and more. However, if you wanted to obtain property testers using this approach in the distribution-free setup, your ordeal might be far from over. The unknown distribution your domain is equipped with can mess with geometric arguments your gridding approach hopes to exploit. This is precisely the setup considered in the paper (i.e, distribution-free testing of function properties).

The essence of downsampling, is captured by a slick proposition that prescribes coarsening as your goto weapon if the
1) fraction of cells on which \(f\) is not constant, and
2) a measure of how non-uniform the unknown distribution D your domain is equipped with is

are both small.

Equipped with this machinery, the paper tackles the task of designing distribution-free testers for boolean monotonicity with the underlying domain being \(\mathbb{R}^d\). The argument is pretty short and improves upon the sample complexity of the corresponding result in the paper by Black-Chakrabarti-Seshadhri. Do check it out, looks like some nice ammo to add to your toolkit.

Let’s stay on board for sublinear time algorithms for gap-edit distance.

Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal by Elazar Goldenberg, Tomasz Kociumaka, Robert Krauthgamer, Barna Saha (arXiv). The edit distance problem needs no introduction. This paper studies the problem of approximating edit distance in sublinear time. In particular, this is formalized by introducing the gapped version of the problem. This entails the following computational task: Fix \(k \in \mathbb{N}\) and some \(c > 0\). Given a pair \(x, y\) of strings over a finite alphabet, decide whether the edit distance between \(x,y\) is at most \(k\) or whether it is at least \(k^c\). This paper resolves the non-adaptive query complexity of edit distance and proves that the above gapped version can be decided in at most \(O\left(\frac{n}{k^{c – 1/2}}\right)\) queries. The paper also proves that this bound is almost optimal up to some polylog factors.

Next up, we have time-optimal algorithms for maximum matching and vertex cover! Sweet, right?

Time-Optimal Sublinear Algorithms for Matching and Vertex Cover by Soheil Behnezhad (arXiv). This paper gives algorithms for the classic problem of estimating the size of vertex cover and maximum matching in graphs in sublinear time (in both adjacency matrix and adjacency list models). This is another nice read for the holiday season — after all the paper obtains time-optimal algorithms (up to polylog factors) in both of these models for a multiplicative-additive \((2, \varepsilon n)\) algorithms. Let me set up some historical context to appreciate the maximum matching result better.

In a classic work, Parnas and Ron gave sublinear time algorithms that obtain a \((2, \varepsilon n)\) estimate to the size of a maximum matching in a graph using query access to the adjacency list in sublinear time. Their sublinear time algorithm is inspired by ideas with roots in distributed algorithms. In particular, their algorithm returns in time
\(\Delta^{O(\log {\frac{\delta}{\varepsilon}})}\) (where \(\Delta = \Delta(G)\) denotes the degree bound) an estimate \(\text{EST}\) to the size of the max-matching where \(OPT \leq \text{EST} \leq 2 \cdot OPT + \varepsilon n\). Unfortunately, algorithms in this Parnas-Ron Framework must necessarily suffer a quasi-polynomial running time because of lower bounds from distributed algorithms. Using a randomized greedy approach to estimating the size of a maximum matching, Yoshida, Yamamoto, and Ito gave the first algorithm which returned a \((2, \varepsilon n)\)-estimate to maximum matching in time \(poly(\Delta/\varepsilon)\). This is where the story essentially froze — though there were results that improved the dependence on \(\Delta\) to \(O(\Delta^2)\). Unfortunately, this is not truly sublinear for large \(\Delta\). In another direction, Kapralov-Mitrovic-Norouzi Fard-Tardos gave an algorithm that estimates the maximum matching size in time \(O(\Delta)\) (which is truly sublinear) but it no longer guarantees a \((2, \varepsilon n)\)-approximation and instead returns a \((O(1), \varepsilon n)\)-approximation. The current paper, as remarked above achieves the best of both worlds.

Final stop, updates on the implicit graph conjecture.

The Implicit Graph Conjecture is False by Hamed Hatami, Pooya Hatami (arXiv). While not a property testing paper per se, it is always good to see an old conjecture resolved one way or the other. In this paper, the Hatami brothers (Pooya and Hamed) refute a three decade old conjecture — the one mentioned in the title. Here is a brief history. Sampath Kannan, Moni Naor and Steven Rudich defined the notion of implicit representation of graphs. Take a family of \(\mathcal{F}\) of graphs and consider a \(n\) vertex graph \(G \in \mathcal{F}\). You say this family admits an efficient implicit representation if there exists an assignment of \(O(\log n)\) length labels to all the vertices in \(V(G)\) such that the adjacencies between every pair of vertices is a function of the labels of the corresponding pair. Crucially, the labeling function may depend on the family, but not on individual graphs in the family. What is cool about families admitting such an efficient implicit representation, is that the number of \(n\)-vertex graphs in this family cannot be any bigger than \(2^{O(n \log n)}\) — that is such families have at most factorial growth rate. Implicit graph conjecture asserts that for every hereditary graph family, the converse also holds. Namely, if the hereditary family has at most factorial growth rate, then the family admits efficient implicit representations. The key, as shown in this paper (which spans all of six pages!) is to choose as your hereditary graph class, the closure of a random collection of graphs. The authors show that now your hand is forced and your representation will no longer be efficient. However, annoyingly, your hereditary graph class need not have a factorial growth rate as taking the closure of a random collection expands to contain all graphs and has growth rate \(2^{\Omega(n^2)}\). The cool thing is, you can avoid this issue by choosing a random collection of slightly sparse random graphs (with \(n^{2-\varepsilon}\) edges). Interestingly, this gives you enough ammo to finally control the growth rate which in turn allows the authors to slay this conjecture.

(EDIT: Omitted Stop, added retroactively).

Sublinear quantum algorithms for estimating von Neumann entropy by Tom Gur, Min-Hsiu Hsieh, Sathyawageeswar Subramanian (arXiv). This paper presents quantum algorithms for the problem of obtaining multiplicative estimates of the Shannon entropy of the familiar classical distributions and the more exotic von Neumann entropy of mixed quantum systems. In particular, the paper presents an \(\widetilde{O}(n^{\frac{1+\eta }{2\gamma^2 }})\)-query quantum algorithm that achieves a \(\gamma\)-multiplicative approximation algorithm for the Shannon Entropy of an input distribution \(\mathbf{p}\) supported on a universe of size \(n\) where \(H(\mathbf{p}) \geq \gamma/\eta\). As if this were not already cool enough, the paper also presents sublinear quantum algorithms for estimating Von Neumann Entropy as well. This is supplemented by a lower bound of \(\Omega(n^{\frac{1}{3 \gamma^2}})\) queries for achieving a \(\gamma\)-factor approximation for any of these two kinds of entropies.

This month saw four papers. One on group testing, another on distribution testing, yet another which makes progress on testing problems on decision trees and the last one on graph property testing. Without further ado, let’s dive in.

Group Testing with Non-identical Infection Probabilities by Mustafa Doger, Sennur Ulukus (arXiv) Consider the classic group testing problem. Here the setup is the following. You are given a bunch of individuals from a population \(\mathcal{P}\). You have an infection vector which records the infection status of each individual in the population where the \(i\)-th individual is infected with probability \(p_i\). You want to recover all the infected individuals. You are allowed to group individuals together and you can test the entire group in a single shot. If the group tests negative, you are happy all the tested individuals are off the hook. Otherwise, if the group tests positive, you need more tests for further classification. This paper proposes a greedy way to build pools of individuals you would test. The pools are built adaptively: as in future pools are built using the knowledge of how the preceding tests fared. The key result in the paper upperbounds the number of tests performed in terms of the entropy of the infection vector.

Uniformity Testing in the Shuffle Model: Simpler, Better, Faster by Clément L. Canonne, Hongyi Lyu (arXiv) Differentially private distribution testing as a research area has been gathering momentum steadily over the last few years. If you read our last month’s post, you might recall there are a wide variety of models of DP each corresponding to a different “threat model”. The most stringent among the most explored models is the “local model”, the least stringent being the “central model” and there is an intermediate threat model, the so called “shuffle model“. This paper simplifies the analysis of uniformity testing algorithm under the shuffle model and presents an algorithm with sample complexity \(O(k^{3/4})\) for testing uniformity over a support of size \([k]\).

On Learning and Testing Decision Tree by Nader H. Bshouty, Catherine A. Haddad-Zaknoon (arXiv) In our December 2020 post, we covered a result of Blanc et al., which proves the following: Suppose you are given a boolean function \(f\) and the property \(\mathcal{P}\) of size-\(s\) decision trees. The result of Blanc et al gives you a function \(g \in \mathcal{P}\) with \(dist(f,g) = O(dist (f, \mathcal{P}))\) where \(g \in \) is guaranteed to have decision tree complexity \(s^{O(\log^2 s)}\). This result implies a bi-criteria tester for the following property: is \(f \in \mathcal{P}\) or is \(f\) \(\varepsilon\)-far from having decision tree complexity \(\phi(s) = s^{O(\log^3 s)}\). The current paper improves this result by presenting a property tester with \(\phi(s) = s^{O(\log^2 s)}\).

The complexity of testing all properties of planar graphs, and the role of isomorphism by Sabyasachi Basu, Akash Kumar, C. Seshadhri (arXiv) (Disclaimer: I am one of the authors of this paper). This paper presents a result that I, in my biased opinion, find interesting. So, here is the setup. You are given a bounded degree planar graph. And I cook up some God-forsaken property and ask you to test it. Turns out, no matter how devilishly I cooked up the property, you can test in with \(\exp(O(\varepsilon^{-2}))\) queries. The nice happenstance is that you also have a matching lower bound of \(\exp(\Omega(\varepsilon^{-2}))\) queries! And interestingly, this lower bound is witnessed by the very natural property of testing isomorphism to a fixed graph which means that isomorphism is the hardest property of planar graphs.