The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal. In this work, we consider the following model of instances: starting with a set of vertices V, a set S ⊆ V of k vertices is chosen and an arbitrary d-regular bipartite graph is added on it; edges between pairs of vertices in S× (V⧵S) and (V⧵S) × (V⧵S) are added with probability p. Since for d = 0, the problem reduces to recovering a planted independent set, we don't expect efficient algorithms for k = o(√n). This problem is a generalization of the planted balanced biclique problem where the bipartite graph induced on S is a complete bipartite graph; [Yevgeny Levanzov, 2018] gave an algorithm for recovering S in this problem when k = Ω(√n). Our main result is an efficient algorithm that recovers (w.h.p.) the planted bipartite graph when k = Ω_p(√{n log n}) for a large range of parameters. Our results also hold for a natural semi-random model of instances, which involve the presence of a monotone adversary. Our proof shows that a natural SDP relaxation for the problem is integral by constructing an appropriate solution to it’s dual formulation. Our main technical contribution is a new approach for construction the dual solution where we calibrate the eigenvectors of the adjacency matrix to be the eigenvectors of the dual matrix. We believe that this approach may have applications to other recovery problems in semi-random models as well. When k = Ω(√n), we give an algorithm for recovering S whose running time is exponential in the number of small eigenvalues in graph induced on S; this algorithm is based on subspace enumeration techniques due to the works of [Alexandra Kolla and Madhur Tulsiani, 2007; Arora et al., 2010; Kolla, 2011].

Consider property testing on bounded degree graphs and let ∊ > 0 denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on ∊. Recent advances in testing minor-freeness have proven that all additive and monotone properties of planar graphs can be tested in poly(∊–1) queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in poly(∊–1) queries? Is there a uniform query complexity upper bound for all planar properties, and what is the “hardest” such property to test? We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in exp(O(∊–2)) queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph requires exp(Ω(∊–2)) queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adaptation of the Newman-Sohler analysis that tracks dependencies on ∊ more carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other. We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in exp(O(∊–1 log ∊–1)) queries. Moreover, testing isomorphism to a fixed forest requires exp(Ω(∊–1)) queries.

Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and$n$be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, one removes a small fraction of edges of the graph controlled by a proximity parameter to get connected components of size independent of$n$. An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a hyperfinite decomposition, without any preprocessing. Given a query vertex v, the partition oracle outputs the component containing v in time independent of n. All the answers are consistent with a single hyperfinite decomposition. The partition oracle of Hassidim et al. runs in time exponential in the proximity parameter per query. They pose the open problem of whether partition oracles which run in time polynomial in reciprocal of proximity parameter can be built. Levi-Ron (ICALP 2013) give a refinement of the previous approach, to get a partition oracle that runs in quasipolynomial time per query. In this paper, we resolve this open problem and give polynomial time partition oracles (in reciprocal of proximity parameter) for bounded degree graphs in any minor-closed family. Unlike the previous line of work based on combinatorial methods, we employ techniques from spectral graph theory. We build on a recent spectral graph theoretical toolkit for minor-closed graph families, introduced by the authors to develop efficient property testers. A consequence of our result is an efficient property tester for any monotone and additive with running time property of minor-closed families (such as bipartite planar graphs). Our result also gives query efficient algorithms for additive approximations for problems such as maximum matching, minimum vertex cover, maximum independent set, and minimum dominating set for these graph families.

Let G be a graph with n vertices and maximum degree d. Fix some minor-closed property P (such as planarity). We say that G is ε-far from P if one has to remove ε dn edges to make it have P. The problem of property testing P was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in ε−1. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in ε−1) query complexity. It is an open problem to get property testers whose query complexity is (dε−1), even for planarity.

Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\mathcal{P}$ (such as planarity). We say that $G$ is $\varepsilon$-far from $\mathcal{P}$ if one has to remove $\varepsilon dn$ edges to make it have $\mathcal{P}$. The problem of property testing $\mathcal{P}$ was introduced in the seminal work of Benjamini, Schramm, and Shapira (Symposium on the Theory of Computing 2008) that gave a tester with query complexity triply exponential in $\varepsilon^{-1}$. Levi and Ron [ ACM Trans. Algorithms, 11 (2005), 24 2015] have given the best tester to date, with a quasi-polynomial (in $\varepsilon^{-1}$) query complexity. It remained an open problem to show whether there is a property tester whose query complexity is $\mathrm{poly}(d\varepsilon^{-1})$, even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity $d\cdot \mathrm{poly}(\varepsilon^{-1})$. The previous line of work on (independent of $n$, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (Foundations of Computer Science 2018) analyzing random walk algorithms that find forbidden minors.

A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone if it flips between 0 and 1 at most $k$ times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. [ Innovations in Theoretical Computer Science, Schloss-Dagstuhl--Leibniz-Zentrum für Informatik GmBH, Wadern, Germany, 2017, 29] initiate the study of $k$-monotone functions in the area of property testing, and Newman et al. [SODA, SIAM, Philadelphia, 2017, pp. 1582--1597] study testability of families characterized by freeness from order patterns on real-valued functions over the line $[n]$ domain. We study $k$-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas, Ron, and Rubinfeld [ J. Comput. System Sci., 72 (2006), pp. 1012--1042]. In this process we show strong lower bounds on testing $k$-monotonicity. Specifically, we show that testing 2-monotonicity on the hypercube nonadaptively with one-sided error requires an exponential in $\sqrt{n}$ number of queries. This behavior shows a stark contrast with testing (1-)monotonicity, which only needs $\tilde{O}\mleft(\sqrt{n}\mright)$ queries. Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being $n^{.01}$-monotone also requires an exponential number of queries.

Let G be an undirected, bounded degree graph with n vertices. Fix a finite graph H, and suppose one must remove ε n edges from G to make it H-minor free (for some small constant ε > 0). We give an n1/2+o(1)-time randomized procedure that, with high probability, finds an H-minor in such a graph. As an application, suppose one must remove ε n edges from a bounded degree graph G to make it planar. This result implies an algorithm, with the same running time, that produces a K3,3or K5minor in G. No prior sublinear time bound was known for this problem. By the graph minor theorem, we get an analogous result for any minor-closed property. Up to no(1)factors, this resolves a conjecture of Benjamini-Schramm-Shapira (STOC 2008) on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal, by an Ω(√n) lower bound of Czumaj et al (RSA 2014). Prior to this work, the only graphs H for which non-trivial one-sided property testers were known for H-minor freeness are the following: H being a forest or a cycle (Czumaj et al, RSA 2014), K2,k, (k× 2)-grid, and the k-circus (Fichtenberger et al, Arxiv 2017).

We consider the problem of recovering a planted pseudorandom 3-coloring in expanding and low threshold-rank graphs. Alon and Kahale [SICOMP 1997] gave a spectral algorithm to recover the coloring for a random graph with a planted random 3-coloring. We show that their analysis can be adapted to work when coloring is pseudorandom i.e., all color classes are of equal size and the size of the intersection of the neighborhood of a random vertex with each color class has small variance. We also extend our results to partial colorings and low threshold-rank graphs to show the following: * For graphs on n vertices with threshold-rank r, for which there exists a 3-coloring that is eps-pseudorandom and properly colors the induced subgraph on (1-gamma)n vertices, we show how to recover the coloring for (1 - O(gamma + eps)) n vertices in time (rn)^{O(r)}. * For expanding graphs on n vertices, which admit a pseudorandom 3-coloring properly coloring all the vertices, we show how to recover such a coloring in polynomial time. Our results are obtained by combining the method of Alon and Kahale, with eigenspace enumeration methods used for solving constraint satisfaction problems on low threshold-rank graphs.

A Boolean $k$-monotone function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical monotone functions, which are the $1$-monotone functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following: - We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; - We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=ω(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\varepsilon})}$ queries, while learning $k$-monotone functions requires $2^{Ω(k\cdot \sqrt d\cdot{1/\varepsilon})}$ queries (Blais et al. (RANDOM 2015)). - We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.