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Register a new userLinear Diophantine Equations, Group CSPs, and Graph Isomorphism
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Martin Grohe
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In recent years, we have seen several approaches to the graph isomorphism problem based on generic mathematical programming or algebraic (Grobner basis) techniques. For most of these, lower bounds have been established. In fact, it has been shown that the pairs of nonisomorphic CFI-graphs (introduced by Cai, Furer, and Immerman in 1992 as hard examples for the combinatorial Weisfeiler-Leman algorithm) cannot be distinguished by these mathematical algorithms. A notable exception were the algebraic algorithms over the field GF(2), for which no lower bound was known. Another, in some way even stronger, approach to graph isomorphism testing is based on solving systems of linear Diophantine equations (that is, linear equations over the integers), which is known to be possible in polynomial time. So far, no lower bounds for this approach were known. Lower bounds for the algebraic algorithms can best be proved in the framework of proof complexity, where they can be phrased as lower bounds for algebraic proof systems such as Nullstellensatz or the (more powerful) polynomial calculus. We give new hard examples for these systems: families of pairs of non-isomorphic graphs that are hard to distinguish by polynomial calculus proofs simultaneously over all prime fields, including GF(2), as well as examples that are hard to distinguish by the systems-of-linear-Diophantine-equations approach. In a previous paper, we observed that the CFI-graphs are closely related to what we call group CSPs: constraint satisfaction problems where the constraints are membership tests in some coset of a subgroup of a cartesian power of a base group (Z_2 in the case of the classical CFI-graphs). Our new examples are also based on group CSPs (for Abelian groups), but here we extend the CSPs by a few non-group constraints to obtain even harder instances for graph isomorphism.
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We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Grobner 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 Grobner 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 different from 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.
Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and Haa stad, there has been a flurry of works on PCSPs [BBKO19,KO19,WZ20]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as the polymorphisms are analyzed. The polymorphisms of PCSPs are much richer than CSPs. In the Boolean case, we still do not know if dichotomy for PCSPs exists analogous to Schaefers dichotomy result for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate $x leq y$. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [BKM21] which is a perfect completeness surrogate of the Unique Games Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every $epsilon>0$, it has polymorphisms where each coordinate has Shapley value at most $epsilon$, else it is NP-hard. The algorithmic part of our dichotomy is based on a structural lemma that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. Of independent interest, we show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.
For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A,B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms $Ato Cto B$, then PCSP(A,B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer $n>1$ and every prime p we give A, B of size n with a single relation of arity $n^p$ such that PCSP(A, B) reduces via a chain of homomorphisms $ Ato Cto B$ to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime $pgeq 7$ we construct A, B of size $p-1$ with a single ternary relation such that PCSP(A, B) reduces via $Ato Cto B$ to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A,B) reduces to tractable CSP(C) for some finite C, then already A or B has tractable CSP. This extends results and answers a question of Deng et al.
Let $F$ be a connected graph with $ell$ vertices. The existence of a subgraph isomorphic to $F$ can be defined in first-order logic with quantifier depth no better than $ell$, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs $K_ell$ and $K_{ell-1}$. We show that, for some $F$, the existence of an $F$ subgraph in emph{sufficiently large} connected graphs is definable with quantifier depth $ell-3$. On the other hand, this is never possible with quantifier depth better than $ell/2$. If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some $F$ be arbitrarily small comparing to $ell$ but never smaller than the treewidth of $F$. Moreover, the definitions over highly connected graphs require quantifier depth strictly more than the density of $F$. Finally, we determine the exact values of these descriptive complexity parameters for all connected pattern graphs $F$ on 4 vertices.
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.
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