On Constraint Satisfaction Problems below P

25th International Workshop/20th Annual Conference of the EACSL

László Egri

Others  :  http://drops.dagstuhl.de/opus/volltexte/2011/3232/pdf/19.pdf
PID  :  44640

来源: CEUR

【 摘 要 】

Symmetric Datalog, a fragment of the logic programming language Datalog, is conjectured to capture all constraint satisfaction problems (CSP) in logarithmic space [10]. Therefore developing tools that help us understand whether or not a CSP can be defined in symmetric Datalog is an important task. A simple, well-known fact is that for any CSP, a fixed set of structures O (an obstruction set) can be defined such that a CSP instance I is a yes-instance iff no structure in O maps homomorphically to I. A CSP having X-duality means that the set O can be chosen to have property X. It is widely known that a CSP is definable in Datalog and linear Datalog iff that CSP has bounded treewidth [12] and bounded pathwidth duality [6], respectively. In the case of symmetric Datalog, Bulatov, Krokhin and Larose ask for such a duality in [4]. We provide two such dualities, and we give applications. In particular, we give a short and simple new proof of the main result of [8] that “Maltsev + Datalog ⇒ symmetric Datalog”.In the second part of the paper, we provide some evidence for the conjecture that every CSP in nondeterministic logarithmic space (NL) is definable in the Datalog fragment linear Datalog [6]. We recall that every problem in NL can be defined by a linear Datalog program with negation and access to an order over the domain of its input (linDat(suc,¬)) [6, 13, 15], or by a poly-size family of nondeterministic branching programs [20]. We consider the following restrictions of the previous models: read-once linDat(suc) (1-linDat(suc)), and monotone read- once nondeterministic branching programs (mnBP1). Although restricted, these models can still define NL-complete problems such as directed st-Connectivity, and also nontrivial problems in NL which are not definable in linear Datalog. We show that any CSP definable by a 1-linDat(suc) program or by a poly-size family of mnBP1s can also be defined by a linear Datalog program. It also follows that a wide class of CSPs–CSPs which do not have bounded pathwidth duality (e.g. the P-complete Horn-3Sat problem)–cannot be defined by any 1-linDat(suc) program or by any poly-size family of mnBP1s.1998 ACM Subject Classification F.1.3 Complexity Measures and Classes