Pebble Games and Linear Equations

Grohe, Martin; Otto, Martin

doi:10.1017/jsl.2015.28

Cited by 55 publications

(61 citation statements)

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“…LP hierarchies and FPC. Another intriguing connection between counting logics and linear programming is established in [Atserias and Maneva 2012;Grohe and Otto 2012] where it is shown that the hierarchy of Sherali-Adams relaxations [Sherali and Adams 1990] of the graph isomorphism integer program interleaves with equivalence in k-variable logic with counting (C k ). It is suggested [Atserias and Maneva 2012] that inexpressibility results for C k could be used to derive integrality gaps for such relaxations.…”

Section: Resultsmentioning

confidence: 99%

Solving Linear Programs without Breaking Abstractions

Anderson

Dawar

Holm

2015

J. ACM

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We show that the ellipsoid method for solving linear programs can be implemented in a way that respects the symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make choices, between variables or constraints in the program unless they are distinguished by properties definable from the program. In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable in FPC. We use this to show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [Blass et al. 1999]. On the way to defining a suitable separation oracle for the maximum matching program, we provide FPC formulas defining canonical maximum flows and minimum cuts in undirected capacitated graphs.

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Section: Resultsmentioning

confidence: 99%

Solving Linear Programs without Breaking Abstractions

Anderson

Dawar

Holm

2015

J. ACM

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“…The following lemma is, at least implicitly, from [15]. As the formal framework is different there, we nevertheless give a proof.…”

Section: Lemma 42mentioning

confidence: 97%

“…Complementing recent research on linear and semidefinite programming approaches to GI [1,10,15,19,20], we investigate the power of GI-algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of all these approaches is to encode isomorphisms between two graphs as solutions to a system of equations and possibly inequalities and then try to solve this system or relaxations of it.…”

Section: Introductionmentioning

confidence: 99%

“…Atserias and Maneva [1] and independently Malkin [19] proved that the Weisfeiler-Lehman algorithm is closely tied to the Sherali-Adams hierarchy [22] of increasingly tighter LP-relaxations of the integer linear program for GI described above: the distinguishing power of k-WL is between that of the (k − 1)st and kth level of the Sherali-Adams hierarchy. Otto and the second author of this paper [15] gave a precise correspondence between k-WL and the nonnegative solutions to a system of linear equations between the (k − 1)st and kth level of the Sherali-Adams hierarchy. Already in 1992, Cai, Fürer, and Immerman [8] had proved that for every k there are non-isomorphic graphs G k , H k (called CFI-graphs in the following) of size O(k) that are not distinguished by k-WL, and combined with the results of Atserias-Maneva and Malkin, this implies that no sublinear level of the Sherali-Adams hierarchy suffices to decide isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Limitations of Algebraic Approaches to Graph Isomorphism Testing

Berkholz

Grohe

2015

Lecture Notes in Computer Science

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We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gröbner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic = 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations.We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus.

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“…1 The GRAPHISOMORPHISM problem can be thought of as kind of constraint satisfaction problem (CSP), and readers familiar with LP/SDP hierarchies for CSPs might see an analogy between k-dimensional WL and level-k LP/SDP relaxations. A very interesting recent work of Atserias and Maneva [AM13] (see also [GO12]) shows that this is more than just an analogy -it shows that the power of WL k is precisely sandwiched between that of the kth and (k+1)st level of the canonical Sherali-Adams LP hierarchy [SA90]. (In fact, it had long been known [RSU94] that WL 1 is equivalent in power to the basic LP relaxation of GISO.)…”

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confidence: 99%

Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs

O’Donnell

Wright

et al. 2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Building on work of Cai, Fürer, and Immerman [CFI92], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n-vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomorphism requires degree Ω(n). In other words, we show an Ω(n)round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 − 10 −14 )-isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are α-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [Fei02] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism problem is hard. I.e. for every > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 − )-isomorphic from pairs which are not even (1 − 0 )-isomorphic for some universal constant 0 . Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest. the heuristic is powerful enough to work correctly for all trees and for almost all n-vertex graphs in the Erdős-Rényi G(n, 1/2) model [BES80, BK79]. (We say the heuristic "works correctly" on a graph G if the stabilized coloring for G is distinct from the stabilized coloring of any graph not isomorphic to G.) To overcome WL's failure for regular graphs, several researchers independently introduced the "k-dimensional generalization" WL k (see [Wei76, CFI92] for discussion). Briefly, in the WL k heuristic, each k-tuple of vertices has a color, and color refinement involves looking at all "neighbors" of each k-tuple of vertices (v 1 , . . . , v k ) (where the neighbors are all tuples of the formThe WL k heuristic can be performed in time n k+O(1) and is thus a polynomial-time algorithm for any constant k.The WL k heuristic is very powerful. For example, it is known to work correctly in polynomial time for all graphs which exclude a fixed minor [Gro12], a class which includes all graphs of bounded treewidth or bounded genus. Spielman's 2 O(n 1/3 ) -time graph isomorphism algorithm [Spi96] for strongly regular graphs is achieved by WL k with k = O(n 1/3 ). The WL k algorithm with k = O( √ n) is also a key component in the 2 O( √ n log n) -time GISO algorithm [BL83]. Throughout the '80s there was some speculation that GISO might be solvable on all graphs by running the WL k algorithm with k = O(log n) or even k = O(1). However this was disproved in the notable work of Cai, Fürer, and Immerman [CFI92], which showed the existence of nonisomorphic n-vertex graphs G and H which are not distinguished by WL k unless k = Ω(n). 1 The GRAPHISOMORPHISM problem can be thought of as kind of constraint satisfaction problem (CSP), and readers familiar with LP/SDP hierarchies for CSPs might see an analogy between k-dimensional WL and level-k LP/SDP relaxations. A very interesting recent work of Atserias and Maneva [AM13] (see also [GO12]) shows tha...

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Pebble Games and Linear Equations

Cited by 55 publications

References 27 publications

Solving Linear Programs without Breaking Abstractions

Solving Linear Programs without Breaking Abstractions

Limitations of Algebraic Approaches to Graph Isomorphism Testing

Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs

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