On Constraint Satisfaction Problems below P

Abstract

Symmetric Datalog, a fragment of the logic programming language Datalog, is conjectured to capture all constraint satisfaction problems (CSP) in L. Therefore developing tools that help us understand whether or not a CSP can be defined in symmetric is an important task. It is widely known that a CSP is definable in and linear iff that CSP has bounded treewidth and bounded pathwidth duality, respectively. In the case of symmetric Datalog, Bulatov, Krokhin and Larose ask for such a duality [2008]. We provide two such dualities, and give applications. In particular, we give a short and simple new proof of the result of Dalmau and Larose that Maltsev + -> symmetric Datalog [2008]. In the second part of the paper, we provide some evidence for the conjecture of Dalmau [2002] that every CSP in NL is definable in linear Datalog. Our results also show 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 polynomial size family of monotone read-once nondeterministic branching programs. We consider the following restrictions of the previous models: read-once linDat(suc) (1-linDat(suc)), and monotone readonce 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 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.