Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer ReviewedPostprint (author's final draft

Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree

Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

We show that feasibility of the t^th level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class ?_t of graphs such that graphs G and H are not distinguished by the t^th level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in ?_t. By analysing the treewidth of graphs in ?_t we prove that the 3t^th level of Sherali-Adams linear programming hierarchy is as strong as the t^th level of Lasserre. Moreover, we show that this is best possible in the sense that 3t cannot be lowered to 3t-1 for any t. The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family ?_t^+ of graphs. Additionally, we give characterisations of level-t Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler-Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the t^th level of the Lasserre hierarchy with non-negativity constraints

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

Definable inapproximability: New challenges for duplicator

AbstractWe consider the hardness of approximation of optimization problems from the point of view of definability. For many $\textrm{NP}$-hard optimization problems it is known that, unless $\textrm{P} = \textrm{NP}$, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.</jats:p

Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer Reviewe

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields