arXiv subjects

Andrei Krokhin

Publications and source records attributed to Andrei Krokhin.

At least 19 recordsLinked to original sources

arXiv2025

Approximating 1-in-3 SAT by linearly ordered hypergraph 3-colouring is NP-hard

Given a satisfiable instance of 1-in-3 SAT, it is NP-hard to find a satisfying assignment for it, but it may be possible to efficiently find a solution subject to a weaker (not necessarily Boolean) predicate than `1-in-3'. There is a folklore conjecture predicting which choices of weaker predicates lead to tractability and for which the task remains \NP-hard. One specific predicate, corresponding to the problem of linearly ordered $3$-colouring of 3-uniform hypergraphs, has been mentioned in several recent papers as an obstacle to further progress in proving this conjecture. We prove that the problem for this predicate is NP-hard, as predicted by the conjecture. We use the Promise CSP framework, where the complexity analysis is performed via the algebraic approach, by studying the structure of polymorphisms, which are multidimensional invariants of the problem at hand. The analysis of polymorphisms is in general a highly non-trivial task, and topological combinatorics was recently discovered to provide a useful tool for this. There are two distinct ways in which it was used: one is based on variations of the Borsuk-Ulam theorem, and the other aims to classify polymorphisms up to certain reconfigurations (homotopy). Our proof, whilst combinatorial in nature, shows that our problem is the first example where the features behind the two uses of topology appear together. Thus, it is likely to be useful in guiding further development of the topological method aimed at classifying Promise CSPs. An easy consequence of our result is the hardness of another specific Promise CSP, which was recently proved by Filakovsk\'y et al. by employing a deep topological analysis of polymorphisms.

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arXiv2024

Functors on relational structures which admit both left and right adjoints

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $Λ$ and $Γ$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $Λ(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $Γ(\mathbf B)$. If this is the case, $Λ$ is called the left adjoint to $Γ$ and $Γ$ the right adjoint to $Λ$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems.

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arXiv2023

1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise

The 1-in-3 and Not-All-Equal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Equal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems.

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arXiv2022

Topology and adjunction in promise constraint satisfaction

The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of a graph that is promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.

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arXiv2022

An invitation to the promise constraint satisfaction problem

The study of the complexity of the constraint satisfaction problem (CSP), centred around the Feder-Vardi Dichotomy Conjecture, has been very prominent in the last two decades. After a long concerted effort and many partial results, the Dichotomy Conjecture has been proved in 2017 independently by Bulatov and Zhuk. At about the same time, a vast generalisation of CSP, called promise CSP, has started to gain prominence. In this survey, we explain the importance of promise CSP and highlight many new very interesting features that the study of promise CSP has brought to light. The complexity classification quest for the promise CSP is wide open, and we argue that, despite the promise CSP being more general, this quest is rather more accessible to a wide range of researchers than the dichotomy-led study of the CSP has been.

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arXiv2019

The complexity of 3-colouring $H$-colourable graphs

We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite graph $H$. In the context of promise constraint satisfaction problems, Brakensiek and Guruswami conjectured that this hardness result extends to promise graph homomorphism as follows: fix any non-bipartite graph $H$ and another graph $G$ with a homomorphism from $H$ to $G$, it is NP-hard to find a homomorphism to $G$ from a given $H$-colourable graph. Arguably, the two most important special cases of this conjecture are when $H$ is fixed to be the complete graph on 3 vertices (and $G$ is any graph with a triangle) and when $G$ is the complete graph on 3 vertices (and $H$ is any 3-colourable graph). The former case is equivalent to the notoriously difficult approximate graph colouring problem. In this paper, we confirm the Brakensiek-Guruswami conjecture for the latter case. Our proofs rely on a novel combination of the universal-algebraic approach to promise constraint satisfaction, that was recently developed by Barto, Bulín and the authors, with some ideas from algebraic topology.

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arXiv2019

Algebraic approach to promise constraint satisfaction

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 20 years. A new version of the CSP, the promise CSP (PCSP) has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms --- high-dimensional symmetries of solution spaces --- to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this paper we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem, and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a "measure of symmetry" that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by improving the state-of-the-art in approximate graph colouring: we show that, for any $k\geq 3$, it is NP-hard to find a $(2k-1)$-colouring of a given $k$-colourable graph.

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arXiv2018

Robust algorithms with polynomial loss for near-unanimity CSPs

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying a $(1-g(\varepsilon))$-fraction of constraints on any $(1-\varepsilon)$-satisfiable instance, where the loss function $g$ is such that $g(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some $g$) have been characterised by Barto and Kozik, with the general bound on the loss $g$ being doubly exponential, specifically $g(\varepsilon)=O((\log\log(1/\varepsilon))/\log(1/\varepsilon))$. It is natural to ask when a better loss can be achieved: in particular, polynomial loss $g(\varepsilon)=O(\varepsilon^{1/k})$ for some constant $k$. In this paper, we consider CSPs with a constraint language having a near-unanimity polymorphism. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter $k$ in the loss depends on the size of the domain and the arity of the relations in $Γ$, while the other works for a special ternary near-unanimity operation called dual discriminator with $k=2$ for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a fixed domain and 2-SAT. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for CSP.

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arXiv2018

Towards a Characterization of Constant-Factor Approximable Finite-Valued CSPs

In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. An instance of VCSP is given by a finite set of variables and a sum of functions belonging to Γ and depending on a subset of the variables. Each function takes values in [0, 1] specifying costs of assignments of labels to its variables, and the goal is to find an assignment of labels to the variables that minimizes the sum. A recent result of Ene et al. says that, under the mild technical condition that Γ contains the equality relation, the basic LP relaxation is optimal for constant-factor approximation for VCSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP(Γ). We also show how these algebraic conditions can in principle be used to round solutions of the basic LP relaxation, and how, for several examples that cover all previously known cases, this leads to efficient constant-factor approximation algorithms. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation.

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arXiv2017

On the hardness of losing weight

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 better (lighter, i.e., having strictly less Hamming weight) solution within a given distance from the initial solution. We classify the complexity, both classical and parameterized, of such problems by a Schaefer-style dichotomy result, that is, with a restricted set of allowed types of constraints. Our results show that there is a considerable amount of such problems that are NP-hard, but fixed-parameter tractable when parameterized by the distance.

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arXiv2017

Binarisation for Valued Constraint Satisfaction Problems

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bulin et al. [LMCS'15] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to Minimum-Cost Homomorphism Problems over a fixed digraph.

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arXiv2017

The Complexity of General-Valued CSPs

An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates infeasible assignments. The goal is to find an assignment of labels to the variables that minimizes the sum. We study (assuming that P $\ne$ NP) how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in $\{0,\infty\}$ corresponds to ordinary CSPs, where one deals only with the feasibility issue and there is no optimization. This case is the subject of the Algebraic CSP Dichotomy Conjecture predicting for which constraint languages CSPs are tractable and for which NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Živný. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs.

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arXiv2016

On Algebras with many symmetric operations

We show that, for each finite algebra A, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by A has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point- free automorphism.

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arXiv2015

A Reduction from Valued CSP to Min Cost Homomorphism Problem for Digraphs

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 numbers or infinity, and is chosen from a fixed finite set of functions called a constraint language. The case when all functions take only values 0 and infinity is known as the constraint satisfaction problem (CSP). It is known that any CSP with fixed constraint language is polynomial-time equivalent to one where the constraint language contains a single binary relation (i.e. a digraph). A recent proof of this by Bulin et al. gives such a reduction that preserves most of the algebraic properties of the constraint language that are known to characterize the complexity of the corresponding CSP. We adapt this proof to the more general setting of VCSP to show that each VCSP with a fixed finite (valued) constraint language is equivalent to one where the constraint language consists of one $\{0,\infty\}$-valued binary function (i.e. a digraph) and one finite-valued unary function, the latter problem known as the (extended) Minimum Cost Homomorphism Problem for digraphs. We also show that our reduction preserves some important algebraic properties of the (valued) constraint language.

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arXiv2013

Oracle Tractability of Skew Bisubmodular Functions

In this paper we consider skew bisubmodular functions as introduced in [9]. We construct a convex extension of a skew bisubmodular function which we call Lovász extension in correspondence to the submodular case. We use this extension to show that skew bisubmodular functions given by an oracle can be minimised in polynomial time.

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arXiv2010

The complexity of the list homomorphism problem for graphs

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

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arXiv2007

Hard constraint satisfaction problems have hard gaps at location 1

An instance of Max CSP is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint languages) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1, i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P = NP. We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P $\neq$ NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. We use these results to partially answer open questions and strengthen results by Engebretsen et al. [Theor. Comput. Sci., 312 (2004), pp. 17--45], Feder et al. [Discrete Math., 307 (2007), pp. 386--392], Krokhin and Larose [Proc. Principles and Practice of Constraint Programming (2005), pp. 388--402], and Jonsson and Krokhin [J. Comput. System Sci., 73 (2007), pp. 691--702]

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arXiv2006

The approximability of MAX CSP with fixed-value constraints

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. In this paper, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x=a), is either solvable exactly in polynomial-time or else is APX-complete, even if the number of occurrences of variables in instances are bounded. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description relies on the well-known algebraic combinatorial property of supermodularity.

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