In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of pos... more In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of positive-weight constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so that the total weight of satisfied constraints is maximized. We consider this problem and its variant Max AW CSP where the weights are allowed to be both positive and negative, and study how the complexity of the problems depends on the allowed constraint types. We prove that Max AW CSP over an arbitrary finite domain exhibits a dichotomy: it is either polynomial-time solvable or NP-hard. Our proof builds on two results that may be of independent interest: one is that the problem of finding a maximum H-colourable subdigraph in a given digraph is either NP-hard or trivial depending on H, and the other a dichotomy result for Max CSP with a single allowed constraint type.
We study the computational complexity of the qualitative algebra which is a temporal constraint f... more We study the computational complexity of the qualitative algebra which is a temporal constraint formalism that combines the point algebra, the point-interval algebra and Allen's interval algebra. We identify all tractable fragments and show that every other fragment is NP-complete.
We study logical techniques for deciding the computational complexity of infinite-domain constrai... more We study logical techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems (CSPs). For the fundamental algebraic structure G</font > = (\mathbb R; L1,L2,...)\Gamma=(\mathbb R; L_1,L_2,\dots) where \mathbb R\mathbb R are the real numbers and L 1,L 2,... is an enumeration of all linear relations with rational coefficients, we prove that a semilinear relation R (i.e., a relation
For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a st... more For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.
In constraint satisfaction problems over finite domains, some variables can be frozen, that is, t... more In constraint satisfaction problems over finite domains, some variables can be frozen, that is, they take the same value in all possible solutions. We study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in the instances. We show that the complexity of such problems is determined by certain algebraic properties of these relations. Under the assumption that NP = coNP (and consequently PTIME = NP), we characterize all tractable problems, and describe large classes of NP-complete, coNP-complete, and DP-complete problems. As an application of these results, we completely classify the complexity of the problem in two cases:
Let Γ be a structure with a finite relational signature and a first-order definition in (R; * , +... more Let Γ be a structure with a finite relational signature and a first-order definition in (R; * , +) with parameters from R, that is, a relational structure over the real numbers where all relations are semi-algebraic sets. In this article, we study the computational complexity of constraint satisfaction problem (CSP) for Γ: the problem to decide whether a given primitive positive sentence is true in Γ. We focus on those structures Γ that contain the relations ≤, {(x, y, z) | x+y = z} and {1}. Hence, all CSPs studied in this article are at least as expressive as the feasibility problem for linear programs. The central concept in our investigation is essential convexity: a relation S is essentially convex if for all a, b ∈ S, there are only finitely many points on the line segment between a and b that are not in S. If Γ contains a relation S that is not essentially convex and this is witnessed by rational points a, b, then we show that the CSP for Γ is NP-hard. Furthermore, we characterize essentially convex relations in logical terms. This different view may open up new ways for identifying tractable classes of semi-algebraic CSPs. For instance, we show that if Γ is a first-order expansion of (R; +, 1, ≤), then the CSP for Γ can be solved in polynomial time if and only if all relations in Γ are essentially convex (unless P=NP).
In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (po... more 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 is 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.
The instances of the Weighted Maximum H-Colourable Subgraph problem (Max H-Col) are edge-weighted... more The instances of the Weighted Maximum H-Colourable Subgraph problem (Max H-Col) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H = K k this problem is equivalent to Max k-cut. Färnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of Max H-Col. Here, we investigate the properties of this parameter on circular complete graphs K p/q , where 2 ≤ p/q ≤ 3. The results are extended to K 4 -minor-free graphs. We also consider connections withŠámal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.
We study techniques for deciding the computational complexity of infinite-domain constraint satis... more We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain fundamental algebraic structures Delta, we prove definability dichotomy theorems of the following form: for every first-order expansion Gamma of Delta, either Gamma has a quantifier-free Horn definition in Delta, or there is an element d of Gamma such that all non-empty relations in Gamma contain
In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of pos... more In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of positive-weight constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so that the total weight of satisfied constraints is maximized. We consider this problem and its variant Max AW CSP where the weights are allowed to be both positive and negative, and study how the complexity of the problems depends on the allowed constraint types. We prove that Max AW CSP over an arbitrary finite domain exhibits a dichotomy: it is either polynomial-time solvable or NP-hard. Our proof builds on two results that may be of independent interest: one is that the problem of finding a maximum H-colourable subdigraph in a given digraph is either NP-hard or trivial depending on H, and the other a dichotomy result for Max CSP with a single allowed constraint type.
We study the computational complexity of the qualitative algebra which is a temporal constraint f... more We study the computational complexity of the qualitative algebra which is a temporal constraint formalism that combines the point algebra, the point-interval algebra and Allen's interval algebra. We identify all tractable fragments and show that every other fragment is NP-complete.
We study logical techniques for deciding the computational complexity of infinite-domain constrai... more We study logical techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems (CSPs). For the fundamental algebraic structure G</font > = (\mathbb R; L1,L2,...)\Gamma=(\mathbb R; L_1,L_2,\dots) where \mathbb R\mathbb R are the real numbers and L 1,L 2,... is an enumeration of all linear relations with rational coefficients, we prove that a semilinear relation R (i.e., a relation
For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a st... more For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.
In constraint satisfaction problems over finite domains, some variables can be frozen, that is, t... more In constraint satisfaction problems over finite domains, some variables can be frozen, that is, they take the same value in all possible solutions. We study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in the instances. We show that the complexity of such problems is determined by certain algebraic properties of these relations. Under the assumption that NP = coNP (and consequently PTIME = NP), we characterize all tractable problems, and describe large classes of NP-complete, coNP-complete, and DP-complete problems. As an application of these results, we completely classify the complexity of the problem in two cases:
Let Γ be a structure with a finite relational signature and a first-order definition in (R; * , +... more Let Γ be a structure with a finite relational signature and a first-order definition in (R; * , +) with parameters from R, that is, a relational structure over the real numbers where all relations are semi-algebraic sets. In this article, we study the computational complexity of constraint satisfaction problem (CSP) for Γ: the problem to decide whether a given primitive positive sentence is true in Γ. We focus on those structures Γ that contain the relations ≤, {(x, y, z) | x+y = z} and {1}. Hence, all CSPs studied in this article are at least as expressive as the feasibility problem for linear programs. The central concept in our investigation is essential convexity: a relation S is essentially convex if for all a, b ∈ S, there are only finitely many points on the line segment between a and b that are not in S. If Γ contains a relation S that is not essentially convex and this is witnessed by rational points a, b, then we show that the CSP for Γ is NP-hard. Furthermore, we characterize essentially convex relations in logical terms. This different view may open up new ways for identifying tractable classes of semi-algebraic CSPs. For instance, we show that if Γ is a first-order expansion of (R; +, 1, ≤), then the CSP for Γ can be solved in polynomial time if and only if all relations in Γ are essentially convex (unless P=NP).
In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of (po... more 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 is 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.
The instances of the Weighted Maximum H-Colourable Subgraph problem (Max H-Col) are edge-weighted... more The instances of the Weighted Maximum H-Colourable Subgraph problem (Max H-Col) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H = K k this problem is equivalent to Max k-cut. Färnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of Max H-Col. Here, we investigate the properties of this parameter on circular complete graphs K p/q , where 2 ≤ p/q ≤ 3. The results are extended to K 4 -minor-free graphs. We also consider connections withŠámal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.
We study techniques for deciding the computational complexity of infinite-domain constraint satis... more We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain fundamental algebraic structures Delta, we prove definability dichotomy theorems of the following form: for every first-order expansion Gamma of Delta, either Gamma has a quantifier-free Horn definition in Delta, or there is an element d of Gamma such that all non-empty relations in Gamma contain
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Papers by Peter Jonsson