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Register a new userThe Descriptive Complexity of Subgraph Isomorphism without Numerics
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Added by
Oleg Verbitsky
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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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.
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Let $v(F)$ denote the number of vertices in a fixed connected pattern graph $F$. We show an infinite family of patterns $F$ such that the existence of a subgraph isomorphic to $F$ is expressible by a first-order sentence of quantifier depth $frac23,v(F)+1$, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any $F$ with using less than $frac23,v(F)-2$ first-order variables.
Given a graph $F$, let $I(F)$ be the class of graphs containing $F$ as an induced subgraph. Let $W[F]$ denote the minimum $k$ such that $I(F)$ is definable in $k$-variable first-order logic. The recognition problem of $I(F)$, known as Induced Subgraph Isomorphism (for the pattern graph $F$), is solvable in time $O(n^{W[F]})$. Motivated by this fact, we are interested in determining or estimating the value of $W[F]$. Using Olarius characterization of paw-free graphs, we show that $I(K_3+e)$ is definable by a first-order sentence of quantifier depth 3, where $K_3+e$ denotes the paw graph. This provides an example of a graph $F$ with $W[F]$ strictly less than the number of vertices in $F$. On the other hand, we prove that $W[F]=4$ for all $F$ on 4 vertices except the paw graph and its complement. If $F$ is a graph on $t$ vertices, we prove a general lower bound $W[F]>(1/2-o(1))t$, where the function in the little-o notation approaches 0 as $t$ inreases. This bound holds true even for a related parameter $W^*[F]le W[F]$, which is defined as the minimum $k$ such that $I(F)$ is definable in the infinitary logic $L^k_{inftyomega}$. We show that $W^*[F]$ can be strictly less than $W[F]$. Specifically, $W^*[P_4]=3$ for $P_4$ being the path graph on 4 vertices. Using the lower bound for $W[F]$, we also obtain a succintness result for existential monadic second-order logic: A usage of just one monadic quantifier sometimes reduces the first-order quantifier depth at a super-recursive rate.
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.
In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of textsc{Clique}, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by $k := text{vc}(G_1)+text{vc}(G_2)$ the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time $2^{O(k log k)}$. We complement this result by showing that, unless the ETH fails, it cannot be solved in time $2^{o(k log k)}$. This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by $k$, unless $NP subseteq mathsf{coNP}/poly$. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.
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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