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We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is…

Computational Complexity · Computer Science 2020-01-17 Miriam Backens , Andrei Bulatov , Leslie Ann Goldberg , Colin McQuillan , Stanislav Živný

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs.…

Data Structures and Algorithms · Computer Science 2023-07-20 Charlie Carlson , Ewan Davies , Alexandra Kolla , Aditya Potukuchi

In a valued constraint satisfaction problem (VCSP), the goal is to find an assignment of labels to variables that minimizes a given sum of functions. Each function in the sum depends on a subset of variables, takes values which are rational…

Computational Complexity · Computer Science 2015-07-08 Robert Powell , Andrei Krokhin

The tractability of certain CSPs for dense or sparse instances is known from the 90s. Recently, the densification and the sparsification of CSPs were formulated as computational tasks and the systematical study of their computational…

Computational Complexity · Computer Science 2022-11-22 Rustem Takhanov

In the Constraint Satisfaction Problem (CSP for short) the goal is to decide the existence of a homomorphism from a given relational structure $G$ to a given relational structure $H$. If the structure $H$ is fixed and $G$ is the only input,…

Logic in Computer Science · Computer Science 2025-10-14 Andrei A. Bulatov , Amirhossein Kazeminia

Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem, and its extension to all…

Quantum Physics · Physics 2025-02-27 Eric Culf , Kieran Mastel

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must…

Data Structures and Algorithms · Computer Science 2020-12-03 Joshua Brakensiek , Venkatesan Guruswami , Marcin Wrochna , Stanislav Živný

We study the complexity of local search for the Boolean constraint satisfaction problem (CSP), in the following form: given a CSP instance, that is, a collection of constraints, and a solution to it, the question is whether there is a…

Data Structures and Algorithms · Computer Science 2017-11-13 Andrei Krokhin , Dániel Marx

Many AI synthesis problems such as planning or scheduling may be modelized as constraint satisfaction problems (CSP). A CSP is typically defined as the problem of finding any consistent labeling for a fixed set of variables satisfying all…

Artificial Intelligence · Computer Science 2013-03-25 Thomas Schiex

We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances. These problems include \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat,…

Computational Complexity · Computer Science 2007-05-23 Harry B. Hunt , Madhav V. Marathe , Venkatesh Radhakrishnan , Richard E. Stearns

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers, which may be seen as variants of CSPs in the mould of quantified CSPs (QCSPs). We show that a single counting quantifier strictly…

Computational Complexity · Computer Science 2011-12-14 Florent Madelaine , Barnaby Martin , Juraj Stacho

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions over a finite domain. An instance of the problem is specified by…

Computational Complexity · Computer Science 2015-03-19 Vladimir Kolmogorov , Stanislav Zivny

The Promise Constraint Satisfaction Problem (PCSP for short) is a generalization of the well-studied Constraint Satisfaction Problem (CSP). The PCSP has its roots in such classic problems as the Approximate Graph Coloring and the…

Computational Complexity · Computer Science 2025-12-08 Arash Beikmohammadi , Andrei A. Bulatov

We give a complexity dichotomy theorem for the counting Constraint Satisfaction Problem (#CSP in short) with complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of complex-valued functions,…

Computational Complexity · Computer Science 2015-03-19 Jin-Yi Cai , Xi Chen

Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is…

Group Theory · Mathematics 2026-04-22 Eric Samperton , Armin Weiß

The constraint satisfaction problem asks to decide if a set of constraints over a relational structure $\mathcal{A}$ is satisfiable (CSP$(\mathcal{A})$). We consider CSP$(\mathcal{A} \cup \mathcal{B})$ where $\mathcal{A}$ is a structure and…

Computational Complexity · Computer Science 2024-08-28 Peter Jonsson , Victor Lagerkvist , George Osipov

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base…

Computational Complexity · Computer Science 2013-10-25 Andrei A. Bulatov , Martin Dyer , Leslie Ann Goldberg , Mark Jerrum , Colin McQuillan

We study a family of problems, called \prob{Maximum Solution}, where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean…

Computational Complexity · Computer Science 2011-11-10 Peter Jonsson , Fredrik Kuivinen , Gustav Nordh

Assigning jobs onto identical machines with the objective to minimize the maximal load is one of the most basic problems in combinatorial optimization. Motivated by product planing and data placement, we study a natural extension called…

Data Structures and Algorithms · Computer Science 2019-09-27 Klaus Jansen , Alexandra Lassota , Marten Maack

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard…

Computational Complexity · Computer Science 2018-04-24 Holger Dell , Eun Jung Kim , Michael Lampis , Valia Mitsou , Tobias Mömke