Benchmark Graphs for Practical Graph Isomorphism

The state-of-the-art solvers for the graph isomorphism problem can readily solve generic instances with tens of thousands of vertices. Indeed, experiments show that on inputs without particular combinatorial structure the algorithms scale almost linearly. In fact, it is non-trivial to create challenging instances for such solvers and the number of difficult benchmark graphs available is quite limited. We describe a construction to efficiently generate small instances for the graph isomorphism problem that are difficult or even infeasible for said solvers. Up to this point the only other available instances posing challenges for isomorphism solvers were certain incidence structures of combinatorial objects (such as projective planes, Hadamard matrices, Latin squares, etc.). Experiments show that starting from 1500 vertices our new instances are several orders of magnitude more difficult on comparable input sizes. More importantly, our method is generic and efficient in the sense that one can quickly create many isomorphism instances on a desired number of vertices. In contrast to this, said combinatorial objects are rare and difficult to generate and with the new construction it is possible to generate an abundance of instances of arbitrary size. Our construction hinges on the multipedes of Gurevich and Shelah and the Cai-F\"{u}rer-Immerman gadgets that realize a certain abelian automorphism group and have repeatedly played a role in the context of graph isomorphism. Exploring limits of such constructions, we also explain that there are group theoretic obstructions to generalizing the construction with non-abelian gadgets.Comment: 32 page

Logarithmic Weisfeiler--Leman and Treewidth

In this paper, we show that the $(3k+4)$-dimensional Weisfeiler--Leman algorithm can identify graphs of treewidth $k$ in $O(\log n)$ rounds. This improves the result of Grohe & Verbitsky (ICALP 2006), who previously established the analogous result for $(4k+3)$-dimensional Weisfeiler--Leman. In light of the equivalence between Weisfeiler--Leman and the logic $\textsf{FO} + \textsf{C}$ (Cai, F\"urer, & Immerman, Combinatorica 1992), we obtain an improvement in the descriptive complexity for graphs of treewidth $k$. Precisely, if $G$ is a graph of treewidth $k$, then there exists a $(3k+5)$-variable formula $\varphi$ in $\textsf{FO} + \textsf{C}$ with quantifier depth $O(\log n)$ that identifies $G$ up to isomorphism

Descriptive complexity of graph spectra.

Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic are cospectral, and this is not the case with , nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.OZ was supported by CONACyT-Mexico Grant 384665, SS was supported by EPSRC and The Royal Society

Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.Comment: 33 pages, 8 Figure

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the Cutting Planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2011 under the name of CP cutwidth. In this paper, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs G and H, we show a direct connection between the Weisfeiler-Leman differentiation number WL(G, H) of the graphs and the width of a CP refutation for the corresponding isomorphism formula Iso(G, H). In particular, we show that if WL(G, H) ? k, then there is a CP refutation of Iso(G, H) with width k, and if WL(G, H) > k, then there are no CP refutations of Iso(G, H) with width k-2. Similar results are known for other proof systems, like Resolution, Sherali-Adams, or Polynomial Calculus. We also obtain polynomial-size CP refutations from our width bound for isomorphism formulas for graphs with constant WL-dimension

Investigating Logics for Feasible Computation

The most celebrated open problem in theoretical computer science is, undoubtedly, the problem of whether P = NP. This is actually one instance of the many unresolved questions in the area of computational complexity. Many different classes of decision problems have been defined in terms of the resources needed to recognize them on various models of computation, such as deterministic or non-deterministic Turing machines, parallel machines and randomized machines. Most of the non-trivial questions concerning the inter-relationship between these classes remain unresolved. On the other hand, these classes have proved to be robustly defined, not only in that they are closed under natural transformations, but many different characterizations have independently defined the same classes. One such alternative approach is that of descriptive complexity, which seeks to define the complexity, not of computing a problem, but of describing it in a language such as the Predicate Calculus. It is particularly interesting that this approach yields a surprisingly close correspondence to computational complexity classes. This provides a natural characterization of many complexity classes that is not tied to a particular machine model of computation

On the Combinatorial and Logical Complexities of Algebraic Structures

In this thesis, we investigate the combinatorial and logical complexities of several algebraic structures, including groups, quasigroups, certain families of strongly regular graphs, and relation algebras. In Chapter 3, we leverage the Weisfeiler&ndash;Leman algorithm for groups (Brachter &amp; Schweitzer, LICS 2020) to improve the parallel complexity of isomorphism testing for several families of groups including (i) coprime extensions H ⋉ N where H is O(1)-generated and N is Abelian (c.f., Qiao, Sarma, &amp; Tang, STACS 2011), (ii) direct product decompositions, and (iii) groups without Abelian normal subgroups (c.f., Babai, Codenotti, &amp; Qiao, ICALP 2012). Furthermore, we show that the weaker count-free Weisfeiler&ndash;Leman algorithm is unable to even identify Abelian groups. As a consequence, we obtain that FO fails to capture all polynomial-time computable queries even on Abelian groups. Nonetheless, we leverage the count-free variant of Weisfeiler&ndash; Leman in tandem with bounded non-determinism and limited counting to obtain a new upper bound of &beta;1MAC0 (FOLL) for isomorphism testing of Abelian groups. This improves upon the previous TC0 (FOLL) upper bound due to Chattopadhyay, Toran, &amp; Wagner (ACM Trans. Comput. Theory, 2013). Weisfeiler&ndash;Leman is equivalent to the first in a hierarchy of Ehrenfeucht&ndash;Fra&uml;ıss&acute;e pebble games (Hella, Ann. Pur. Appl. Log., 1989). In Chapter 4, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fra&uml;ıss&acute;e bijective pebble game in Hella&rsquo;s (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht-Fra&uml;ıss&acute;e bijective pebble game in Hella&rsquo;s hierarchy.&nbsp; Our main result is that, in the pebble game characterization, only O(1) pebbles and O(1) rounds are sufficient to identify all groups without Abelian normal subgroups. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella&rsquo;s results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only O(1) variables and O(1) quantifier depth.&nbsp; In Chapter 5, we show that Graph Isomorphism (GI) is not AC0 -reducible to several problems, including the Latin Square Isotopy problem and isomorphism testing of several families of Steiner designs. As a corollary, we obtain that GI is not AC0 -reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner 2-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in &beta;2FOLL, which cannot compute Parity (Chattopadhyay, Tor&acute;an, &amp; Wagner, ibid.). Finally, in Chapter 6, we shed new light on the spectrum of the relation algebra we call An, which is obtained by splitting the non-flexible diversity atom of 67 into n symmetric atoms. Precisely, we show that the minimum value in Spec(An) is at most 2n6+o(1), which is the first polynomial bound and improves upon the previous bound due to Dodd &amp; Hirsch (J. Relat. Methods Comput. Sci. 2013). We also improve the lower bound to 2n2 + Ω(n&radic;logn). Prior to the work in this thesis, only the trivial bound of n2 + 2n + 3 was known.</p