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

Abstract

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. LP, SDP or Groebner-based relaxations are now known to stay proper relaxations of isomorphism, their relative strength, besides the obvious relationships, was not fully understood. Since SDP is a proper generalization of LP, one may be tempted to guess that the Lasserre SDP hierarchy could perhaps distinguish more graphs than its LP sibling. Interestingly, we prove this not to be the case: for the isomorphism problem, the strength of the Lasserre hierarchy collapses to that of the Sherali-Adams hierarchy, up to a small loss in the level of the hierarchy.Concretely, we show that there exist a constant c ≥ 1 such that if two given graphs are distinguishable in the k-th level of the Lasserre hierarchy, then they must also be distinguishable in the ck-th level of the Sherali-Adams hierarchy. The constant c loss comes from the number of variables for expressing the SDP exact feasibility problem in the bounded-variable counting logic. It should be noted that our proof is indirect as it relies on the correspondance between indistinguishability in k-variable counting logic and the k-th level Sherali-Adams relaxation of graph isomorphism [3]. The question whether the collapse can be shown to hold directly, by lifting LP-feasible solutions into SDP-feasible ones, remains an interesting one.This collapse result has some curious consequences. For one it says that, for distinguishing graphs, the spectral methods that underlie the Lasserre hierarchy are already available in low levels of the Sherali-Adams hierarchy. This may sound surprising, but aligns well with the known fact that indistinguishability by 3-variable counting logic captures graph spectra [7], together with the abovementioned correspondance between k-variable counting logic and the k-th level of the Sherali-Adams hierarchy.By moving to the duals, our results can be read in terms of Sums-of-Squares (SOS) and Sherali-Adams (SA) proofs, and used to get consequences for Polynomial Calculus (PC) proofs as a side bonus. In terms of proofs, we show that if there is a degree-k SOS proof that two graphs are not isomorphic, then there is also a degree-ck SA proof. In turn, it was already known from before, by combining the results in [3] and [5], that if there is a degree-ck SA proof then there is also a degre...

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

“…To prove Condition (iii), let u * be a child of t Let us first deal with the common elements t i = u i for i ∈ [3]. Note that min * (t i ) t * u * .…”

“…If there is a vertex s ∈ B such that there is a unique connected component C of G \ {s} with B ⊆ {s} ∪ V (C), then there are vertices b j i , s k for i ∈ identifies G, where iso G,(b j i ,s k |i∈ [4],j∈ [3],k∈ [6]) is the formula from Lemma 19. Otherwise, let s ∈ B be a vertex such that G \ {s} has multiple connected components C i and let G i := G[V (C i ) ∪ {s}].…”

“…The algorithm has striking connections to numerous areas of mathematics and computer science, which surely is a reason why research on it has been active since its introduction over half a century ago. For example, there are tight connections to linear and semidefinite programming [2,3,20], homomorphism counting [8,10], and the algebra of coherent configurations [6]. Most recently, the WL algorithm has been applied in several interesting machine-learning contexts [1,16,33,34,39].…”

The graph isomorphism problem

“…1 forms a singleton color class with respect to χ and G is 3-connected. Now let c 1 , c 2 , c 3 ∈ N H (C ′ ) be distinct and also let v i ∈ χ −1 (c i ) for i ∈ [3]. Also let T be a spanning tree of H…”

“…First note that this gives an algorithm for computing the automorphism group of G. Since G is t-CR-bounded there is some i ∞ ≥ 0 such that P i∞ is the discrete partition. Hence, Aut(G) ≤ Γ i∞ 3 where Γ i∞ is a Γ t -group and the automorphism group of G can be computed using Theorems 4.19 and 4.20.…”