News for July 2016

After a few slow months, property testing is back with a bang! This month we have a wide range on results ranging from classic problems like junta testing to new models of property testing.

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism, by Eric Blais, ClĂ©ment L. Canonne, Talya Eden, Amit Levi, and Dana Ron (arXiv). The problem of junta testing requires little introduction. Given a boolean function \(f:\{-1,+1\}^n \mapsto \{-1,+1\}\), a \(k\)-junta only depends on \(k\) of the input variables. A classic problem has been that of property testing \(k\)-juntas, and a rich line of work (almost) resolves the complexity to be \(\Theta(k/\epsilon)\). But what of tolerant testing, where we want to tester to accept functions close to being a junta? Previous work has shown that existing testers are tolerant, but with extremely weak parameters. This paper proves: there is a \(poly(k/\epsilon)\)-query tester that accepts every function \(\epsilon/16\)-close to a \(k\)-junta and rejects every function \(\epsilon\)-far from being a \(4k\)-junta. Note that the “gap” between the junta classes is a factor of 4, and this is the first result to achieve a constant gap. The paper also gives testers when we wish to reject functions far from being a \(k\)-junta (exactly matching the definition of tolerant testng), but the tester has an exponential dependence on \(k\). The results have intriguing connections with isomorphism testing, and there is a neat use of constrained submodular minimization.

Testing Pattern-Freeness, by Simon Korman and Daniel Reichman (arXiv). Consider a string \(I\) of length \(n\) and a “pattern” \(J\) of length \(k\), both over some alphabet \(\Sigma\). Our aim is to property test if \(I\) is \(J\)-free, meaning that \(I\) does not contains \(J\) as a substring. This problem can also be generalized to the 2-dimensional setting, where \(I\) and \(J\) are matrices with entries in \(\Sigma\). This formulation is relevant in image testing, where we need to search for a template pattern in a large image. The main results show that these testing problems can be solved in time \(O(1/\epsilon)\), with an intriguing caveat. If the pattern \(J\) has at least \(3\) distinct symbols, the result holds. If \(J\) is truly binary, then \(J\) is not allowed to be in a specified set of forbidden patterns. The main tool is a modification lemma that shows how to “kill” a specified occurrence of \(J\) without introducing a new one. This lemma is not true for the forbidden patterns, resulting in the dichotomy of the results.

Erasure-Resilient Property Testing, by Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma (arXiv). Property testing begins with a query model for \(f: \mathcal{D} \mapsto \mathcal{R}\), so we can access any \(f(x)\). But what if an adversary corrupted some fraction of the input? Consider monotonicity testing on the line, so \(f: [n] \mapsto \mathbb{R}\). We wish to test if \(f\) is \(\epsilon\)-far from being monotone, but the adversary corrupts/hides an \(\alpha\)-fraction of the values. When we query such a position, we receive a null value. We wish to accept if there exists some “filling” of the corrupted values that makes \(f\) monotone, and reject if all “fillings” keep \(f\) far from monotone. It turns out the standard set of property testers are not erasure resilient, and fail on this problem. This papers gives erasure resilient testers for properties like monotonicity, Lipschitz continuity, and convexity. The heart of the techniques involves a randomized binary tree tester for the line that can avoid the corrupted points.

Partial Sublinear Time Approximation and Inapproximation for Maximum Coverage, by Bin Fu (arXiv). Consider the classic maximum coverage problem. We have \(m\) sets \(A_1, A_2, \ldots, A_m\) and wish to pick the \(k\) of them with the largest union size. The query model allows for membership queries, cardinality queries, and generation of random elements from a set. Note that the size of the input can be thought of as \(\sum_i |A_i|\). Can one approximate this problem without reading the full input? This paper gives a \((1-1/e)\)-factor approximation that runs in time \(poly(k)m\log m\), and is thus sublinear in the input size. The algorithm essentially implements a greedy approximation algorithm in sublinear time. It is shown the the linear dependence on \(m\) is necessary: there is no constant factor approximation that runs in \(q(n) m^{1-\delta}\) (where \(n\) denotes the maximum cardinality, \(q(\cdot)\) is an arbitrary function, and \(\delta > 0\)).

Local Testing for Membership in Lattices, by Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, and Elena Grigorescu (arXiv). Inspired by the theory of locally testable codes, this paper introduces local testing in lattices. Given a set of basis vectors \(b_1, b_2, \ldots\) in \(\mathbb{Z}^n\), the lattice \(L\) is the set of all integer linear combinations of the basic vectors. This is the natural analogue of linear error correcting codes (where everything is done over finite fields). Given some input \(t \in \mathbb{Z}^n\), we wish to determine if \(t \in L\), or is far (defined through some norm) from all vectors in \(L\), by querying a constant number of coordinates in \(t\). We assume that \(L\) is fixed, so it can be preprocessed arbitrarily. This opens up a rich source of questions, and this work might be seen as only the first step in this direction. The papers shows a number of results. Firstly, there is a family of “code formula” lattices for which testers exists (with almost matching lower bounds). Furthermore, with high probability over random lattices, testers do not exist. Analogous to testing codes, if a constant query lattice tester exists, then there exists a canonical constant query tester.

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