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

Proof Complexity Meets Algebra

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, 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 bounded 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 Sums-of-Squares refutations of sublinear degree, a fact for which we provide an alternative proof. We hence 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 bounded-degree Lovasz-Schrijver satisfies both requirements

On the power of symmetric linear programs

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991) showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.Peer ReviewedPostprint (author's final draft

Homomorphism Problems for First-Order Definable Structures

We investigate several variants of the homomorphism problem: given two relational structures, is there a homomorphism from one to the other? The input structures are possibly infinite, but definable by first-order interpretations in a fixed structure. Their signatures can be either finite or infinite but definable. The homomorphisms can be either arbitrary, or definable with parameters, or definable without parameters. For each of these variants, we determine its decidability status

Agregacja sądów a agregacja preferencji

In the paper we present an introduction to the theory of judgment aggregation and discuss its relation to the theory of preference aggregation. We compare the formal model of judgment aggregation, based on logic, with the formal model of preference aggregation. Finally, we present a theorem in judgmentaggregation which is an exact analogue of Arrow's theorem for strict preferences.Artykuł stanowi wprowadzenie do teorii agregacji sądów oraz porusza temat jej związków z teorią agregacji preferencji. Oparty na logice model formalny agregacji sądów porównany jest z modelem formalnym agregacji preferencji. Przedstawiony zostaje ponadto wynik w teorii agregacji sądów stanowiący dokładny odpowiednik twierdzenia Arrowa dla mocnych porządków

JUDGMENT AGGREGATION AND PREFERENCE AGGREGATION

In the paper we present an introduction to the theory of judgment aggregation and discuss its relation to the theory of preference aggregation. We compare the formal model of judgment aggregation, based on logic, with the formal model of preference aggregation. Finally, we present a theorem in judgmentaggregation which is an exact analogue of Arrow's theorem for strict preferences

Extended constraint satisfaction problems

To solve an instance of the constraint satisfaction problem (CSP) one has to find an assignment of values to variables that satisfies given constraints. This thesis concerns two different extensions of the constraint satisfaction problem.The first extension allows for an infinite number of constraints in an instance. It is phrased in the language of sets with atoms, which provides finite means of representation -- an instance is founded upon a fixed infinite relational structure, and defined by finitely many first order formulas. We prove decidability for this so-called locally finite CSP, and establish tight complexity bounds in the special case when the set of possible values is finite.We further use the constraint satisfaction framework to analyse the computational model of Turing machines with atoms (TMAs), whose alphabet, state space, and transition relation are orbit-finite sets with atoms (usually infinite but finitely presentable). We give an effective characterisation of those alphabets for which TMAs determinise, with applications to descriptive complexity.The second extension of the CSP that we consider is known as the valued constraint satisfaction problem (VCSP). It provides a common framework for many discrete optimisation problems. We use algebraic tools to establish a necessary condition for tractability of VCSPs parametrised by sets of allowed types of constraints. We conjecture that our condition is also sufficient, and verify whether the conjecture agrees with all previously known results.Aby rozwiązać daną instancję problemu spełnialności więzów, należy znaleźć takie przypisanie wartości do zmiennych, żeby spełnione były wszystkie więzy. Ta rozprawa dotyczy dwóch rozszerzeń problemu spełnialności więzów.Pierwsze rozszerzenie dopuszcza nieskończenie wiele więzów w instancji. Jest ono sformalizowane w języku teorii zbiorów z atomami, który pozwala na skończoną reprezentację -- dla ustalonej nieskończonej struktury relacyjnej każda instancja zadana jest przez skończenie wiele formuł logiki pierwszego rzędu. Dowodzimy, że ten problem, który nazywamy lokalnie skończonym problemem spełnialności więzów, jest rozstrzygalny. Podajemy także ścisłe oszacowanie jego złożoności obliczeniowej w szczególnym przypadku, gdy liczba możliwych war- tości jest skończona.Następnie stosujemy narzędzia teorii spełnialności więzów do analizy mo- delu obliczeniowego maszyn Turinga z atomami, których alfabet, zbiór stanów, oraz relacja przejścia są orbitowo-skończonymi zbiorami z atomami (są one zazwyczaj nieskończone, ale możliwa jest ich skończona reprezentacja). Uzyskujemy efektywną charakteryzację tych alfabetów, dla których maszyny Turinga z atomami się determinizują, a ponadto pokazujemy zastosowanie tego wyniku do teorii złożoności opisowej. Drugie rozpatrywane przez nas rozszerzenie problemu spełnialności więzów jest znane jako problem spełnialności więzów z wartościami. Pozwala ono na formalizację wielu dyskretnych problemów optymalizacyjnych. Używając narzędzi algebraicznych ustanawiamy warunek konieczny, aby problem spełnialności wię- zów z wartościami, sparametryzowany przez ustalony zbiór dopuszczalnych typów więzów, był rozwiązywalny w czasie wielomianowym. Formułujemy ponadto hipotezę, która głosi, że podany przez nas warunek jest jednoczeście dostateczny i sprawdzamy jej zgodność ze wszystkimi znanymi wcześniej wynikami

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