Verifying proofs in constant depth

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's

The isomorphism conjecture for constant depth reductions

For any class C closed under TC0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under u-uniform AC0 reductions are isomorphic under u-uniform AC0-computable isomorphisms

Dynamic Complexity Meets Parameterised Algorithms

Dynamic Complexity studies the maintainability of queries with logical formulas in a setting where the underlying structure or database changes over time. Most often, these formulas are from first-order logic, giving rise to the dynamic complexity class DynFO. This paper investigates extensions of DynFO in the spirit of parameterised algorithms. In this setting structures come with a parameter k and the extensions allow additional "space" of size f(k) (in the form of an additional structure of this size) or additional time f(k) (in the form of iterations of formulas) or both. The resulting classes are compared with their non-dynamic counterparts and other classes. The main part of the paper explores the applicability of methods for parameterised algorithms to this setting through case studies for various well-known parameterised problems

On the Descriptive Complexity of Color Coding

Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing - purely in terms of the syntactic structure of describing formulas - when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in fpt just because of the way they are commonly described using logical formulas

An Atypical Survey of Typical-Case Heuristic Algorithms

Heuristic approaches often do so well that they seem to pretty much always give the right answer. How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences? This article first discusses how a series of results by Berman, Buhrman, Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the early 1970s through the early 1990s, explicitly or implicitly limited how well heuristic algorithms can do on NP-hard problems. In particular, many desirable levels of heuristic success cannot be obtained unless severe, highly unlikely complexity class collapses occur. Second, we survey work initiated by Goldreich and Wigderson, who showed how under plausible assumptions deterministic heuristics for randomized computation can achieve a very high frequency of correctness. Finally, we consider formal ways in which theory can help explain the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012 issue of SIGACT New

On the cubical geometry of Higman's group

We investigate the cocompact action of Higman's group on a CAT(0) square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman's group, and show that the group is both hopfian and co-hopfian. We actually prove a stronger rigidity result about the endomorphisms of Higman's group: Every non-trivial morphism from the group to itself is an automorphism. We also study the geometry of the action and prove a surprising result: Although the CAT(0) square complex acted upon contains uncountably many flats, the Higman group does not contain subgroups isomorphic to Z^2. Finally, we show that this action possesses features reminiscent of negative curvature, which we use to prove a refined version of the Tits alternative for Higman's group.Comment: Accepted versio

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