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Asaf Shapira

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23 papers
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23

SODA Conference 2022 Conference Paper

Counting Homomorphic Cycles in Degenerate Graphs

  • Lior Gishboliner
  • Yevgeny Levanzov
  • Asaf Shapira
  • Raphael Yuster

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e. g. , planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al. , which highlighted the importance of the task of counting homomorphic copies of cycles (i. e. , cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2 k and C 2 k +1 in n -vertex graphs of bounded degeneracy in time, where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time. Conversely, one can transform any algorithm for computing the number of homomorphic copies of C 2 k or of C 2 k +1 in n -vertex graphs of bounded degeneracy, into an time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

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STOC Conference 2019 Conference Paper

Testing graphs against an unknown distribution

  • Lior Gishboliner
  • Asaf Shapira

The classical model of graph property testing, introduced by Goldreich, Goldwasser and Ron, assumes that the algorithm can obtain uniformly distributed vertices from the input graph. Goldreich introduced a more general model, called the Vertex-Distribution-Free model (or VDF for short) in which the testing algorithm obtains vertices drawn from an arbitrary and unknown distribution. The main motivation for this investigation is that it can allow one to give different weight/importance to different parts of the input graph, as well as handle situations where one cannot obtain uniformly selected vertices from the input. Goldreich proved that any property which is testable in this model must (essentially) be hereditary, and that several hereditary properties can indeed be tested in this model. He further asked which properties are testable in this model. In this paper we completely solve Goldreich’s problem by giving a precise characterization of the graph properties that are testable in the VDF model. Somewhat surprisingly this characterization takes the following clean form: say that a graph property P is extendable if given any graph G satisfying P , one can add one more vertex to G , and connect it to some of the vertices of G in a way that the resulting graph satisfies P . Then a property P is testable in the VDF model if and only if P is hereditary and extendable.

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SODA Conference 2015 Conference Paper

Decomposing a Graph Into Expanding Subgraphs

  • Guy Moshkovitz
  • Asaf Shapira

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Two examples of our results are the following: Motivated by the Unique Games Conjecture, Trevisan [FOCS O5] and Arora, Barak and Steurer [FOCS 10] showed that given a graph G, one can remove only 1% of G's edges and thus obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are (essentially) best possible even when one is allowed to remove 99% of G's edges. In particular, our results imply that the eigenspace enumeration approach of Arora-Barak-Steurer cannot give (even quasi-) polynomial time algorithms for unique games. A classical result of Lipton, Rose and Tarjan from 1979 states that if ℱ is a hereditary family of graphs and every graph in ℱ has a vertex separator of size n/(log n ) 1+ o (1), then every graph in ℱ has O ( n ) edges. We construct a hereditary family of graphs with vertex separators of size n /(log n ) 1– o (1) such that not all graphs in the family have O ( n ) edges. The above results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

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FOCS Conference 2010 Conference Paper

A Unified Framework for Testing Linear-Invariant Properties

  • Arnab Bhattacharyya 0001
  • Elena Grigorescu
  • Asaf Shapira

There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F 2 -linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan. We obtain the following results: 1. We show that every linear-invariant property that can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations can be tested with one-sided error. 2. We show that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of systems of linear equations. We conjecture that our result from item (1) can be extended to cover systems of linear equations. We further show that the validity of this conjecture would have the following implications: 1. It would imply that every linear-invariant property that is closed under restrictions to linear subspaces is testable with one-sided error. Such a result would unify several previous results on testing Boolean functions, such as the testability of low-degree polynomials and of Fourier dimensionality. 2. It would imply that a linear-invariant property P is testable with one-sided error if and only if P is closed under restrictions to linear subspaces, thus resolving Sudan's problem.

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FOCS Conference 2007 Conference Paper

Approximate Hypergraph Partitioning and Applications

  • Eldar Fischer
  • Arie Matsliah
  • Asaf Shapira

We show that any partition-problem of hypergraphs has an O(n) time approximate partitioning algorithm and an efficient property tester. This extends the results of Goldreich, Goldwasser and Ron who obtained similar algorithms for the special case of graph partition problems in their seminal paper (1998). The partitioning algorithm is used to obtain the following results: ldr We derive a surprisingly simple O(n) time algorithmic version of Szemeredi's regularity lemma. Unlike all the previous approaches for this problem which only guaranteed to find partitions of tower-size, our algorithm will find a small regular partition in the case that one exists; ldr For any r ges 3, we give an O(n) time randomized algorithm for constructing regular partitions of r-uniform hypergraphs, thus improving the previous O(n 2r-1 ) time (deterministic) algorithms. The property testing algorithm is used to unify several previous results, and to obtain the partition densities for the above problems (rather than the partitions themselves) using only poly(1/isin) queries and constant running time.

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FOCS Conference 2005 Conference Paper

A Characterization of the (natural) Graph Properties Testable with One-Sided Error

  • Noga Alon
  • Asaf Shapira

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property-testing. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious one-sided error tester, if and only if P is (semi) hereditary. We stress that any "natural" property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the "natural" graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of Goldreich et al. , [1998] about testing k-colorability, the characterization of Goldreich and Trevisan [2001] of the graph-partition problems that are testable with 1-sided error, the induced vertex colorability properties of Alon et al. , [2000], the induced edge colorability properties of Fischer [2001], a transformation from 2-sided to 1-sided error testing [Goldreich and Trevisan, 2001], as well as a recent result about testing monotone graph properties [Alon and Shapira, 2005]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being perfect, chordal, interval, comparability and more. None of these properties was previously known to be testable.

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FOCS Conference 2005 Conference Paper

Additive Approximation for Edge-Deletion Problems

  • Noga Alon
  • Asaf Shapira
  • Benny Sudakov

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

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STOC Conference 2005 Conference Paper

Every monotone graph property is testable

  • Noga Alon
  • Asaf Shapira

A graph property is called monotone if it is closed under taking (not necessarily induced) subgraphs (or, equivalently, if it is closed under removal of edges and vertices). Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on ε. This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi's Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing.As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P , any graph that is ε -far from satisfying P , contains a subgraph of size depending on ε only, which does not satisfy P . Finally, we prove the following compactness statement: If a graph G is ε-far from satisfying a (possibly infinite) set of graph properties P , then it is at least δ P ε-far from satisfying one of the properties.

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