Min CSP on Four Elements: Moving beyond Submodularity

Abstract

We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. This is unsatisfactory, since such results are interesting from a combinatorial optimisation perspective; there are deep connections with, for instance, submodular and bisubmodular minimisation. We consider the Min and Max CSP problems (i.e. where the cost functions only attain values in {0, 1}) over four-element domains and identify all tractable fragments. Similar classifications were previously known for two-and three-element domains. In the process, we introduce a new class of tractable VCSPs based on a generalisation of submodularity. We also extend and modify a graph-based technique by Kolmogorov and Živný (originally introduced by Takhanov) for efficiently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which otherwise are fairly common when studying the complexity and approximability of VCSPs.) The hardness results are further simplified by the introduction of powerful reduction techniques.

The complexity of finite-valued CSPs

“…Our results generalise all previous partial classifications of finite-valued constraint languages: the classifications of {0, 1}-valued constraint languages on two-element, three-element, and fourelement domains obtained in [22,23], [46], and [49], respectively; the classification of {0, 1}-valued constraint languages containing all unary functions obtained in [26]; the classifications of finitevalued constraint languages on two-element and three-element domains obtained in [15] and [42], re-spectively; the classification of finite-valued constraint languages containing all {0, 1}-valued unary functions obtained in [57]; and the classification of Min-0-Ext problems obtained in [40].…”

“…In all of these classifications, the hardness reductions essentially came from the condition (MC) and tractable cases were characterised by certain specific binary symmetric fractional polymorphisms including the concepts of submodularity [15,26,46], skew bisubmodularity [42], 1-defect [49], and others [40].…”

On Some Combinatorial Optimization Problems : Algorithms and Complexity

“…Their computational complexity was described for languages on the Boolean domain in [27] (see also [28]), on the three-element domain in [55], and on the four-element domain in [57]. The complexity for so-called conservative languages was studied in [31].…”

Robust Satisfiability for CSPs

“…The language-based direction is considerably more active than the structural one, and there are many partial language-based complexity classification results, e.g. [Barto and Kozik 2009;Bulatov 2006;2011;Creignou et al 2001;Deineko et al 2008;Jonsson et al 2006;Jonsson et al 2011], but many central questions are still open.…”

“…There is a line of research aimed at identifying tractable optimization problems CSP(Γ), i.e. those where an optimal assignment can always be found in polynomial time [Cohen et al 2005], and this property is known to be quite restrictive [Deineko et al 2008;Jonsson et al 2006;Jonsson et al 2011]. The following natural notion of tractability, which is stronger than classical tractability of CSP(Γ), but much less restrictive than tractability of optimization version of CSP(Γ), was suggested in [Zwick 1998].…”