Tractable Triangles and Cross-Free Convexity in Discrete Optimisation

Abstract

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.Comment: arXiv admin note: text overlap with arXiv:1008.4035 by other author

The Complexity of Valued Constraint Satisfaction Problems

“…(In fact, the above two theorems hold in a slightly stronger setting; see [82] for more details.) Remark 9.5 Note that the NP-hardness result in Theorem 9.14 requires assignment sets of size up to 3.…”

“…Chapter 8, based on [264], presents recent results on the power of linear programming for valued constraint satisfaction problems. Chapter 9, based on [80] and [82] (preliminary versions [78,79,81]), surveys known results on hybrid tractability. Finally, Chap.…”

“…The joint-winner property [80] for Max-CSPs gives The following classes have been shown to be tractable by Cooper and Živný [82]: A {0,1} ({<, >}), A {0,1} ({>, 0}), and A {0,1} ({>, 1}) (via maximum matching in graphs [105]). …”

“…In this case, the complexity classifications coincide with the classifications of finite-valued VCSPs obtained in Sect. 9.2.3: tractable cases remain tractable (see [82] for more details) and intractable cases remain intractable. …”

“…In this section, we will generalise JWP to the cross-free convexity property (CFC) [82]. First we need to define some notation.…”