In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023), and obtain the following results:

Larrauri and Živný [ICALP’24/ACM ToCL’24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid \({N}\) assuming that a solution exists over a monoid \({M}\) , where both monoids are finite and \({M}\) admits a homomorphism to \({N}\) . Using the algebraic approach to promise constraint satisfaction problems, we extend their complexity classification in two directions: we obtain a complexity dichotomy in the case where arbitrary relations are added to the monoids, and we moreover allow the monoid \({M}\) to be finitely generated.

A multiset of literals, called a clause, is \emph{strongly satisfied} by an assignment if \emph{no} literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies $ρ$ of the clauses, an assignment in which at least $ρ$ of the clauses are \emph{weakly satisfied}, in the sense that an \emph{even} number of literals evaluate to false. In particular, this implies an efficient algorithm for finding an undirected cut of value $ρ$ in a graph $G$ given that a directed cut of value $ρ$ in $G$ is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of $G$ with $ρ$ edges under the same promise.

The Mermin-Peres magic square is a celebrated example of a system of Boolean linear equations that is not (classically) satisfiable but is satisfiable via linear operators on a Hilbert space of dimension four. A natural question is then, for what kind of problems such a phenomenon occurs? Atserias, Kolaitis, and Severini answered this question for all Boolean Constraint Satisfaction Problems (CSPs): For 0-Valid-SAT, 1-Valid-SAT, 2-SAT, Horn-SAT, and Dual Horn-SAT, classical satisfiability and operator satisfiability is the same and thus there is no gap; for all other Boolean CSPs, these notions differ as there are gaps, i.e., there are unsatisfiable instances that are satisfiable via operators on Hilbert spaces. We generalize their result to CSPs on arbitrary finite domains and give an almost complete classification: First, we show that NP-hard CSPs admit a separation between classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces. Second, we show that tractable CSPs of bounded width have no satisfiability gaps of any kind. Finally, we show that tractable CSPs of unbounded width can simulate, in a satisfiability-gap-preserving fashion, linear equations over an Abelian group of prime order $p$; for such CSPs, we obtain a separation of classical satisfiability and satisfiability via operators on infinite-dimensional Hilbert spaces. Furthermore, if $p=2$, such CSPs also have gaps separating classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces.

A celebrated result of Håstad established that, for any constant ε > 0, it is NP-hard to find an assignment satisfying a (1/|G|+ε)-fraction of the constraints of a given 3-LIN instance over an Abelian group G even if one is promised that an assignment satisfying a (1-ε)-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

Fix two non-empty loopless graphs $G$ and $H$ such that $G$ maps homomorphically to $H$. The Maximum Promise Constraint Satisfaction Problem parameterised by $G$ and $H$ is the following computational problem, denoted by MaxPCSP($G$, $H$): Given an input (multi)graph $X$ that admits a map to $G$ preserving a $ρ$-fraction of the edges, find a map from $X$ to $H$ that preserves a $ρ$-fraction of the edges. As our main result, we give a complete classification of this problem under Khot's Unique Games Conjecture: The only tractable cases are when $G$ is bipartite and $H$ contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph $X$ which contains a bipartite subgraph with $ρ$ edges, what is the largest triangle-free subgraph of $X$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 ρ$ edges, thus improving on the subgraph with $0.878 ρ$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.

Abstract. We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework to establish an unconditional lower bound against the semidefinite programming + linear equations model by showing that the relaxation does not solve the approximate graph homomorphism problem and thus, in particular, the approximate graph coloring problem.

Abstract. We show that approximate graph coloring is not solved by the lift-and-project hierarchy for the combination of linear programming and linear Diophantine equations. The proof is based on combinatorial tensor theory.

We study the complexity of approximating the number of answers to a small query \(\varphi\) in a large database \(\mathcal{D}\) . We establish an exhaustive classification into tractable and intractable cases if \(\varphi\) is a conjunctive query possibly including disequalities and negations: — If there is a constant bound on the arity of \(\varphi\) , and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of \(\varphi\) is bounded. — If the arity is unbounded and \(\varphi\) does not have negations, then the problem has an FPTRAS if and only if the adaptive width of \(\varphi\) (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless \(\mathrm{NP}=\mathrm{RP}\) , there is no FPRAS even if the treewidth (and the adaptive width) is \(1\) . However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).

The 1 -in- 3 and N ot -A ll -E qual satisfiability problems for Boolean CNF formulas are two well-known NP -hard problems. In contrast, the promise 1 -in- 3 vs . N ot -A ll -E qual 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.

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.

A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours [FSTTCS'24]. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement.

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.

Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued promise CSPs.

The 1-in-3 and Not-All-Eqal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Eqal 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.

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework to establish an unconditional lower bound against the semidefinite programming + linear equations model, by showing that the relaxation does not solve the approximate graph homomorphism problem and thus, in particular, the approximate graph colouring problem.

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ (C) provides as input a UCQ Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ (C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ 1 ∨ ... ∨ φ l is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φ l . We show that under natural closure properties of C, the problem #UCQ (C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables --- if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ (C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D| d ) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.

The complexity of the promise constraint satisfaction problem \(\operatorname{(PCSP)}(\mathbf{A},\mathbf{B})\) is largely unknown, even for symmetric \(\mathbf{A}\) and \(\mathbf{B}\) , except for the case when \(\mathbf{A}\) and \(\mathbf{B}\) are Boolean. First, we establish a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) where \(\mathbf{A},\mathbf{B}\) are symmetric, \(\mathbf{B}\) is functional (i.e., any \(r-1\) elements of an \(r\) -ary tuple uniquely determines the last one), and \((\mathbf{A},\mathbf{B})\) satisfies technical conditions we introduce called dependency and additivity . This result implies a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) with \(\mathbf{A},\mathbf{B}\) symmetric and \(\mathbf{B}\) functional if (i) \(\mathbf{A}\) is Boolean, or (ii) \(\mathbf{A}\) is a hypergraph of a small uniformity, or (iii) \(\mathbf{A}\) has a relation \(R^{\mathbf{A}}\) of arity at least three such that the hypergraph diameter of \((A,R^{\mathbf{A}})\) is at most one. Second, we show that for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) , where \(\mathbf{A}\) and \(\mathbf{B}\) contain a single relation, \(\mathbf{A}\) satisfies a technical condition called balancedness , and \(\mathbf{B}\) is arbitrary, the combined basic linear programming relaxation and the affine integer programming (AIP) relaxation is no more powerful than the (in general strictly weaker) \({{\rm AIP}}\) relaxation. Balanced \(\mathbf{A}\) include symmetric \(\mathbf{A}\) or, more generally, \(\mathbf{A}\) preserved by a transitive permutation group.

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Živný [SIDMA’20]. Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P :{ 0,1} k → { 0,1} of a fixed arity k , we show that CSP( P ) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP( P ) admits an additive sparsifier for any predicate P : D k → { 0,1} of a fixed arity k on an arbitrary finite domain D .

General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions. In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.

As our first result, we establish a dichotomy for promise constraint satisfaction problems of the form PCSP(A, B), where A is Boolean and symmetric and B is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in B determine the last element. This includes PCSPs of the form PCSP(q-in-r, B), where B is functional, thus making progress towards a classification of PCSP(1-in-3, B), which were studied by Barto, Battistelli, and Berg [STACS’21] for B on three-element domains.As our second result, we show that for PCSP(A, B), where A contains a single symmetric relation and B is arbitrary (and thus not necessarily functional), the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) of Brakensiek et al. [SICOMP’20] is no more powerful than the (in general strictly weaker) AIP relaxation of Brakensiek and Guruswami [SICOMP’21].

We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the “base level” of the hierarchy, and a geometric part - which we call tensorisation - inspired by multilinear algebra. We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, in order to analyse the Sum-of-Squares SDP hierarchy we also characterise the solvability of the standard SDP relaxation through a new minion. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. This work was also supported by UKRI EP/X024431/1. † The full version of the paper can be accessed at https://arxiv.org/abs/2207.02277.

We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. This work was also supported by UKRI EP/X024431/1. † The full version of the paper can be accessed at https://arxiv.org/abs/2210.08293

We show that approximate graph colouring is not solved by constantly many levels of the lift-and-project hierarchy for the combined basic linear programming and affine integer programming relaxation. The proof involves a construction of tensors whose fixed-dimensional projections are equal up to reflection and satisfy a sparsity condition, which may be of independent interest.

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

Abstract. We propose a new algorithm for Promise Constraint Satisfaction Problems (PCSPs). It is a combination of the Constraint Basic LP relaxation and the Affine IP relaxation (CLAP). We give a characterization of the power of CLAP in terms of a minion homomorphism. Using this characterization, we identify a certain weak notion of symmetry which, if satisfied by infinitely many polymorphisms of PCSPs, guarantees tractability. We demonstrate that there are PCSPs solved by CLAP that are not solved by any of the existing algorithms for PCSPs; in particular, not by the [Formula: see text] algorithm of Brakensiek et al. [SIAM J. Comput., 49 (2020), pp. 1232--1248] and not by a reduction to tractable finite-domain CSPs.

Abstract. The approximate graph coloring problem, whose complexity is unresolved in most cases, concerns finding a [Formula: see text]-coloring of a graph that is promised to be [Formula: see text]-colorable, where [Formula: see text]. This problem naturally generalizes 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 analyze 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 coloring and promise graph homomorphism problems.

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes. Baker’s approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA’21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general “sparsity” condition for which a PTAS is known. We extend these results to general-valued CSPs, which include “crisp” (or “strict”) constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element that is at least as feasible as all the others (but possibly less valuable). For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes—a definition by Dvořák [SODA’20] that encompasses all classes where Baker’s technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability —a property of relational structures connected to logics, statistical physics, and random CSPs.

A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns an integer from $\{1, \ldots, k \}$ to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for $r=3$, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on $3$-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO $2$-colouring, one can find in polynomial time an LO $k$-colouring with $k=O(\sqrt[3]{n \log \log n / \log n})$. Second, given an $r$-uniform hypergraph that admits an LO $2$-colouring, we establish NP-hardness of finding an LO $k$-colouring for every constant uniformity $r\geq k+2$. In fact, we determine relationships between polymorphism minions for all uniformities $r\geq 3$, which reveals a key difference between $r

We study the complexity of approximating the number of answers to a small query φ in a large database D. We establish an exhaustive classification into tractable and intractable cases if φ is a conjunctive query possibly including disequalities and negations: - If there is a constant bound on the arity of φ, and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of φ is bounded. - If the arity is unbounded and φ does not have negations, then the problem has an FPTRAS if and only if the adaptive width of φ (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless NP=RP, there is no FPRAS even if the treewidth (and the adaptive width) is 1. However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).

We propose a new algorithm for Promise Constraint Satisfaction Problems (PCSPs). It is a combination of the Constraint Basic LP relaxation and the Affine IP relaxation (CLAP). We give a characterisation of the power of CLAP in terms of a minion homomorphism. Using this characterisation, we identify a certain weak notion of symmetry which, if satisfied by infinitely many polymorphisms of PCSPs, guarantees tractability. We demonstrate that there are PCSPs solved by CLAP that are not solved by any of the existing algorithms for PCSPs; in particular, not by the BLP + AIP algorithm of Brakensiek and Guruswami [SODA'20] and not by a reduction to tractable finite-domain CSPs.

1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) constraint satisfaction problems (CSPs). While both problems are NP-hard, Brakensiek and Guruswami showed [SICOMP'21] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the promise CSP template (1-in-3,NAE) is tractable. Unlike previously established dichotomy results for fragments of promise CSPs (PCSPs), we focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (t-in-k,NAE) by either adding tuples to t-in-k or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by one of the strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek, Guruswami, Wrochna, and Živný [SICOMP'20]. For the latter, we classify all templates as either tractable or NP-hard.

The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand-side structures, the results of Dalmau, Kolaitis, and Vardi [ Proceedings of the 8 th International Conference on Principles and Practice of Constraint Programming, Springer, New York, 2002, pp. 310--326], Grohe [ J. ACM, 54 (2007), 1], and Atserias, Bulatov, and Dalmau [ Proceedings of the 34 th International Colloquium on Automata, Languages and Programming, Springer, New York, 2007, pp. 279--290] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalization of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand-side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$th level of the Sherali--Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognizing the tractable cases; the latter has an application in database theory.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every vertex of \( \mathrm{H}_i \) to every vertex of \( \mathrm{H}_j \) if and only if \( i\lt j \) . We prove that if \( \mathrm{H} \) has both its initial and final strongly connected component (possibly equal) of size 1, then \( \mathrm{QCSP}(\mathrm{H}) \) is in \( \mathsf {NL} \) and otherwise \( \mathrm{QCSP}(\mathrm{H}) \) is \( \mathsf {NP} \) -hard.

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20]. Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{0,1}^k → {0,1} of a fixed arity k, we show that CSP(P)} admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P)} admits an additive sparsifier for any predicate P:D^k → {0,1} of a fixed arity k on an arbitrary finite domain D.

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes.Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable).For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes - a definition by Dvořák [SODA'20] which encompasses all classes where Baker's technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability - a property of relational structures connected to logics, statistical physics, and random CSPs.

We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph H and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class ⊕P of parity problems. We verify their conjecture for all graphs H that exclude the complete graph on 4 vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the ⊕P-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph H. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Göbel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2. A full version of our paper, containing all proofs, is available at https://arxiv.org/abs/2006.16632v2. Here we number key lemmas to match the numbering in the full version.

We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs parametrised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing PTASes. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. The full version [59] containing detailed proofs is available at https://arxiv.org/abs/1911.03204.

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ Theory Comput., 11 (2015), pp. 35--57] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on four vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus{P}$-complete cases, assuming the randomized exponential time hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress toward resolving the conjecture (Faben and Jerrum [ Theory Comput., 11 (2015), pp. 35--57]; Göbel, Goldberg, and Richerby [ ACM Trans. Comput. Theory, 6 (2014), 17; ACM Trans. Comput. Theory, 8 (2016), 12]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2.

A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H . In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved.

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-SAT is NP-hard". They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains.

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring k-colourable graphs with colours for every k ≥ 4. This improves the result of Bulín, Krokhin, and Opršal [STOC'19], who gave NP-hardness of colouring k-colourable graphs with 2k – 1 colours for k ≥ 3, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring k-colourable graphs with colours for sufficiently large k. Thus, for k ≥ 4, we improve from known linear/sub-exponential gaps to exponential gaps. Second, we show that the topology of the box complex of H alone determines whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Opršal [FOCS’19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, non-bipartite) G. We use this technique to establish NP-hardness of H-colouring of G-colourable graphs for H that include but go beyond K3, including square-free graphs and circular cliques (leaving K4 and larger cliques open). Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems. The full version [55] containing detailed proofs is available at https://arxiv.org/abs/1907.00872.

In the field of constraint satisfaction problems (CSPs), 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 distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem-which has recently seen exciting progress [Bulín, Krokhin, and Oprs̆al, Proceedings of the Symposium on Theory of Computing, 2019, pp. 602--613 and Wrochna and Živný, Proceedings of the Symposium on Discrete Algorithms, 2020, pp. 1426--1435] benefiting from a systematic algebraic approach to promise CSPs based on “polymorphisms,” operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [Brakensiek and Guruswami, Proceedings of the Symposium on Discrete Algorithms, 2019, pp. 436--455]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power.

A cut $\varepsilon$-sparsifier of a weighted graph $G$ is a reweighted subgraph of $G$ of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of $\varepsilon$. Since their introduction by Benczúr and Karger [ Approximating s-t minimum cuts in O͂($n^2$) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC'96), 1996, pp. 47--55], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [ SIAM J. Discrete Math., 31 (2017), pp. 1263--1276] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β -acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and β -acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as β -hypertreewidth.

The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β -acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and β -acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as β -hypertreewidth.

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP($\mathcal{C}$,-), in which the goal is, given a relational structure $\mathbf{A}$ from a class $\mathcal{C}$ of structures and an arbitrary structure $\mathbf{B}$, to find the number of homomorphisms from $\mathbf{A}$ to $\mathbf{B}$. Flum and Grohe showed that #CSP($\mathcal{C}$,-) is solvable in polynomial time if $\mathcal{C}$ has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if $\mathcal{C}$ is a recursively enumerable class of relational structures of bounded arity, then assuming FPT $\neq$ #W[1], there are no other cases of #CSP($\mathcal{C}$,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04]. We show that, assuming FPT $\neq$ W[1] (under randomised parametrised reductions) and for $\mathcal{C}$ satisfying certain general conditions, #CSP($\mathcal{C}$,-) is not solvable even approximately for $\mathcal{C}$ of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP($\mathcal{C}$,-). In particular, our condition generalises the case when $\mathcal{C}$ is closed under taking minors.

Let G be a graph that contains an induced subgraph H. A retraction from G to H is a homomorphism from G to H that is the identity function on H. Retractions are very well-studied: Given H, the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this problem is tractable (assuming P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to trees. The result is as follows: (1) Approximately counting retractions to a tree H is in FP if H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if H is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph — a problem which is complete in the approximate counting complexity class RHπ1. (3) Finally, if none of these hold, then approximately counting retractions to H is #P-complete under approximation-preserving reductions. Our second contribution is to locate the retraction counting problem in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms — whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of these problems. The full version containing detailed proofs is available at https://arxiv.org/abs/1807.00590v1 (version from 2 July 2018). The theorem numbering here matches the full version.

A cut epsilon-sparsifier of a weighted graph G is a re-weighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of epsilon. Since their introduction by Benczúr and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting. In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

A homomorphism from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to the vertices of $H$ that preserves edges. A homomorphism is surjective if it uses all of the vertices of $H$, and it is a compaction if it uses all of the vertices of $H$ and all of the nonloop edges of $H$. Hell and Nešetřil gave a complete characterization of the complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$. A complete characterization is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterization of the complexity of counting homomorphisms from an input graph $G$ to a fixed graph $H$. In this paper, we give a complete characterization of the complexity of counting surjective homomorphisms from an input graph $G$ to a fixed graph $H$, and we also give a complete characterization of the complexity of counting compactions from an input graph $G$ to a fixed graph $H$. In an addendum we use our characterizations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper and Živný classified the tractability of binary VCSP instances according to the concept of “triangle,” and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa, Murota, and Živný made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two quadratic M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this article, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the k-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell and Nešetřil gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H.

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions.An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper-Zivny classified the tractability of binary VCSP instances according to the concept of "triangle," and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa-Murota-Zivny made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this paper, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the “bounded width condition”, i.e., it contains weak near-unanimity operations of all arities.

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with the objective function given as a sum of fixed-arity functions; the values are rational numbers or infinity. In Boolean surjective VCSPs variables take on labels from D={0,1} and an optimal assignment is required to use both labels from D. A classic example is the global min-cut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for {0,infinity}-valued constraint languages corresponding to CSPs) obtained by Creignou and Hebrard, and the dichotomy for {0,1}-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.

In this paper we extend the classical notion of strong and weak backdoor sets for SAT and CSP by allowing that different instantiations of the backdoor variables result in instances that belong to different base classes; the union of the base classes forms a heterogeneous base class. Backdoor sets to heterogeneous base classes can be much smaller than backdoor sets to homogeneous ones, hence they are much more desirable but possibly harder to find. We draw a detailed complexity landscape for the problem of detecting strong and weak backdoor sets into heterogeneous base classes for SAT and CSP.

Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic sub-instances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of simple partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes.

We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) to optimality. The condition is that of bounded width, which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-known valued constraint languages are tractable via the third level of the Sherali--Adams relaxation. For the known languages, this is a significantly simpler algorithm than those previously obtained. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Živný [ J. ACM, 60 (2013), 10], and also a dichotomy theorem for the exact solvability of minimum-solution problems. These are generalizations of minimum-ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [ SIAM J. Comput., 30 (2001), pp. 1863--1920], Jonsson, Kuivinen, and Nordh [ SIAM J. Comput., 38 (2008), pp. 329--365], and Uppman [ Proceedings of ICALP'13, Springer, Berlin, 2013, pp. 804--815].

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 Bulín et al. [ Log. Methods Comput. Sci., 11 (2015)] 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.

Characterising tractable fragments of the constraint satisfaction problem (CSP) is an important challenge in theoretical computer science and artificial intelligence. Forbidding patterns (generic sub-instances) provides a means of defining CSP fragments which are neither exclusively language-based nor exclusively structure-based. It is known that the class of binary CSP instances in which the broken-triangle pattern (BTP) does not occur, a class which includes all tree-structured instances, are decided by arc consistency (AC), a ubiquitous reduction operation in constraint solvers. We provide a characterisation of simple partially-ordered forbidden patterns which have this AC-solvability property. It turns out that BTP is just one of five such AC-solvable patterns. The four other patterns allow us to exhibit new tractable classes.

We study the computational complexity of planar valued constraint satisfaction problems (VCSPs). First, we show that intractable Boolean VCSPs have to be self-complementary to be tractable in the planar setting, thus extending a corresponding result of Dvorak and Kupec [ICALP'15] from CSPs to VCSPs. Second, we give a complete complexity classification of conservative planar VCSPs on arbitrary finite domains. As it turns out, in this case planarity does not lead to any new tractable cases, and thus our classification is a sharpening of the classification of conservative VCSPs by Kolmogorov and Zivny [JACM'13].

We study the computational complexity of exact minimization of rational-valued discrete functions. Let Γ be a set of rational-valued functions on a fixed finite domain; such a set is called afinite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimizing a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability forallfinite-valued constraint languages defined on domains ofarbitraryfinite size. We show that every constraint language Γ either admits a binary symmetric fractional polymorphism, in which case the basic linear programming relaxation solves any instance of VCSP(Γ) exactly, or Γ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to VCSP(Γ).

We consider the maximization problem in the value oracle model of functions defined on k -tuples of sets that are submodular in every orthant and r -wise monotone, where k ⩾ 2 and 1 ⩽ r ⩽ k . We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r ). For r = k , we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1/(1+√ k /2. In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP . Extending a result of Ando et al., we show that for any k ⩾ 3, submodularity in every orthant and pairwise monotonicity (i.e., r = 2) precisely characterize k -submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k ) for the maximization problem of k -submodular functions.

Constraint programming is a natural paradigm for many combinatorial optimisation problems. The complexity of constraint satisfaction for various forms of constraints has been widely-studied, both to inform the choice of appropriate algorithms, and to understand better the boundary between polynomial-time complexity and NP-hardness. In constraint programming it is well-known that any constraint satisfaction problem can be converted to an equivalent binary problem using the so-called dual encoding. Using this standard approach any fixed collection of constraints, of arbitrary arity, can be converted to an equivalent set of constraints of arity at most two. Here we show that this transformation, although it changes the domain of the constraints, preserves all the relevant algebraic properties that determine the complexity. Moreover, we show that the dual encoding preserves many of the key algorithmic properties of the original instance. We also show that this remains true for more general valued constraint languages, where constraints may assign different cost values to different assignments. Hence, we obtain a simple proof of the fact that to classify the computational complexity of all valued constraint languages it suffices to classify only binary valued constraint languages.

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrized by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimization problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. For example, if $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to 3-SAT. More generally, if $\Gamma$ contains only $\{0,\infty\}$-valued functions, VCSP($\Gamma$) corresponds to CSP($\Gamma$). If $D=\{0,1\}$ and $\Gamma$ contains all ternary $\{0,1\}$-valued functions, VCSP($\Gamma$) corresponds to Min-3-SAT, in which the goal is to minimize the number of unsatisfied clauses in a 3-CNF instance. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree submodular on arbitrary trees.

The connection between constraint languages and clone theory has been a fruitful line of research on the complexity of constraint satisfaction problems. In a recent result, Cohen et al. [SIAM J. Comput., 42 (2013), pp. 915--1939] have characterized a Galois connection between valued constraint languages and so-called weighted clones. In this paper, we study the structure of weighted clones. We extend the results of Creed and Živný from [Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming, 2011, pp. 210--224] on types of weightings necessarily contained in every nontrivial weighted clone. This result has immediate computational complexity consequences as it provides necessary conditions for tractability of weighted clones and thus valued constraint languages. We demonstrate that some of the necessary conditions are also sufficient for tractability, while others are provably not.

Backdoor sets represent clever reasoning shortcuts through the search space for SAT and CSP. By instantiating the backdoor variables one reduces the given instance to several easy instances that belong to a tractable class.The overall time needed to solve the instance is exponential in the size of the backdoor set, hence it is a challenging problem to find a small backdoor set if one exists; over the last years this problem has been subject of intensive research. In this paper we extend the classical notion of a strong backdoor set by allowing that different instantiations of the backdoor variables result in instances that belong to different base classes; the union of the base classes forms a heterogeneous base class. Backdoor sets to heterogeneous base classes can be much smaller than backdoor sets to homogeneous ones, hence they are much more desirable but possibly harder to find. We draw a detailed complexity landscape for the problem of detecting strong backdoor sets into heterogeneous base classes for SAT and CSP. We provide algorithms that establish fixed-parameter tractability under natural parameterizations, and we contrast the tractability results with hardness results that pinpoint the theoretical limits. Our results apply to the current state-of-the-art of tractable classes of CSP and SAT that are definable by restricting the constraint language.

Submodular functions play a key role in combinatorial optimization and in the study of valued constraint satisfaction problems. Recently, there has been interest in the class of bisubmodular functions, which assign values to disjoint pairs of sets. Like submodular functions, bisubmodular functions can be minimized exactly in polynomial time and exhibit the property of diminishing returns common to many problems in operations research. Recently, the class of k-submodular functions has been proposed. These functions generalize the notion of submodularity to k-tuples of sets, with submodular and bisubmodular functions corresponding to k = 1 and 2, respectively. In this paper, we consider the problem of maximizing bisubmodular and, more generally, k-submodular functions in the value oracle model. We provide the first approximation guarantees for maximizing a general bisubmodular or k-submodular function. We give an analysis of the naive random algorithm as well as a randomized greedy algorithm inspired by the recent randomized greedy algorithm of Buchbinder et al. [FOCS'12] for unconstrained submodular maximization. We show that this algorithm approximates any k-submodular function to a factor of . In the case of bisubmodular functions, our randomized greedy algorithm gives an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible in both the value query model, and under the assumption that NP ≠ RP. Our analysis provides further intuition for the algorithm of Buchbinder et al. [FOCS'12] in the submodular case. Additionally, we show that the naive random algorithm gives a 1/4-approximation for bisubmodular functions, corresponding again to known performance guarantees for submodular functions. Thus, bisubmodular functions exhibit approximability identical to submodular functions in all of the algorithmic contexts we consider.

Let Γ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability for all finite-valued languages defined on domains of arbitrary finite size.

We study the complexity of valued constraint satisfaction problems (VCSPs) parametrized by a constraint language , a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is well understood, see Raghavendra [2008]. However, there is no characterization of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimization perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only {0,∞}-valued cost functions (i.e., relations), such languages have been called conservative and studied by Bulatov [2003, 2011] and recently by Barto [2011]. Since we study valued languages, we call a language conservative if it contains all finite-valued unary cost functions. The computational complexity of conservative valued languages has been studied by Cohen et al. [2006] for languages over Boolean domains, by Deineko et al. [2008] for {0,1}-valued languages (a.k.a Max-CSP), and by Takhanov [2010a] for {0,∞}-valued languages containing all finite-valued unary cost functions (a.k.a. Min-Cost-Hom). We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimorphisms ), then any instance can be solved in polynomial time (via a new algorithm developed in this article), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains are significantly harder than the Boolean cases. The polynomial-time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of Cohen et al. [2008]. Our results generalize previous results by Takhanov [2010a] and (a subset of results) by Cohen et al. [2006] and Deineko et al. [2008]. Moreover, our results do not rely on any computer-assisted search as in Deineko et al. [2008], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.

Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimization problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterized. These results provide a new approach to studying the complexity of discrete optimization. We use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the Boolean case.

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) sub modular on arbitrary lattices, (2) bisubmodular (also known as k-sub modular) on arbitrary finite domains, (3) weakly (and hence strongly) tree-sub modular on arbitrary trees.

We study the complexity of valued constraint satisfaction problems (VCSP) A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is well-understood, see Raghavendra [FOCS'08]. However, there is no characterisation of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only {0, ∞}-valued cost functions (i.e. relations) such languages have been called conservative and studied by Bulatov [LICS'03] and recently by Barto [LICS'11]. Since we study valued languages, we call a language conservative if it contains all finite-valued unary cost functions. The computational complexity of conservative valued languages has been studied by Cohen et al. [AIJ'06] for languages over Boolean domains, by Deineko et al. [JACM'08] for {0, 1}-valued languages (a.k.a Max-CSP), and by Takhanov [STACS'10] for {0, ∞} valued languages containing all finite-valued unary cost functions (a.k.a. Min-Cost-Hom). We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimorphisms), then any instance can be solved in polynomial time (via a new algorithm developed in this paper), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains is significantly harder than the Boolean case. The polynomial-time algorithm we present for the tractable cases is a generalisation of the submodular minimisation problem and a result of Cohen et al. [TCS'08]. Our results generalise previous results by Takhanov [STACS'10] and (a subset of results) by Cohen et al. [AIJ'06] and Deineko et al. [JACM'08]. Moreover, our results do not rely on any computer-assisted search as in Deineko et al. [JACM'08], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.

The minimisation problem of a sum of unary and pairwise functions of discrete variables is a general NP-hard problem with wide applications such as computing MAP configurations in Markov Random Fields (MRF), minimising Gibbs energy, or solving binary Valued Constraint Satisfaction Problems (VCSPs). We study the computational complexity of classes of discrete optimisation problems given by allowing only certain types of costs in every triangle of variable-value assignments to three distinct variables. We show that for several computational problems, the only non- trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property. Our results, apart from giving complete classifications in the studied cases, provide guidance in the search for hybrid tractable classes; that is, classes of problems that are not captured by restrictions on the functions (such as submodularity) or the structure of the problem graph (such as bounded treewidth). Furthermore, we introduce a class of problems with convex cardinality functions on cross-free sets of assignments. We prove that while imposing only one of the two conditions renders the problem NP-hard, the conjunction of the two gives rise to a novel tractable class satisfying the cross-free convexity property, which generalises the joint-winner property to problems of unbounded arity.