Abstract. The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP languages to weighted algebras. We show that the difficulty of VCSP depends only on the weighted variety generated by the associated weighted algebra.Paralleling the results for CSPs we exhibit a reduction to cores and rigid cores which allows us to focus on idempotent weighted varieties. Further, we propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the hardness direction and verify that it agrees with known results for VCSPs on two-element sets [Cohen et al. 2006

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semi-algebraic proof systems, the classical constructions of ppinterpretability, homomorphic equivalence and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterised algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of constant width, or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with refutations of sublinear degree in Sums-of-Squares and Polynomial Calculus over the real-field, for which we provide alternative proofs. We then ask for the existence of a natural proof system with good behaviour with respect to reductions and simultaneously small size refutations beyond bounded width. We give an example of such a proof system by showing that boundeddegree Lovász-Schrijver satisfies both requirements. Finally, building on the known lower bounds, we demonstrate the applicability of the method of reducibilities and construct new explicit hard instances of the graph 3-coloring problem for all studied proof systems.

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...

We study deterministic computability over sets with atoms. We characterize those alphabets for which Turing machines with atoms determinize. To this end, the determinization problem is expressed as a Constraint Satisfaction Problem, and a characterization is obtained from deep results in CSP theory. As an application to Descriptive Complexity Theory, within a substantial class of relational structures including Cai-Fürer-Immerman graphs, we precisely characterize those subclasses where the logic IFP+C captures order-invariant polynomial time computation.