Oliver Kullmann’s research while affiliated with Swansea University and other places

Continuous evolution and improvement of GPGPUs has significantly broadened areas of application. The massively parallel platform they offer, paired with the high efficiency of performing certain operations, opens many questions on the development of suitable techniques and algorithms. In this work, we present a novel algorithm and create a massively parallel, GPGPU‐based solver for enumerating solutions of the N‐Queens problem. We discuss two implementations of our algorithm for GPGPUs and provide insights on the optimizations we applied. We also evaluate the performance of our approach and compare our work to existing literature, showing a clear reduction in computational time.

We introduce the global conflict graph of DQCNFs (dependency quantified conjunctive normal forms), recording clashes between clauses on such universal variables on which all existential variables depend (called “global variables”). The biclique covers of this graph correspond to the eligible clause-slices of the DQCNF which consider only the global variables. We show that all such slices yield satisfiability-equivalent variations. This opens the possibility to realise this slice using as few global variables as possible. We give basic theoretical results and first supporting experimental data. KeywordsQBF solvingDQBF2QCNFbiclique cover problemconflict graphpreprocessingHorn clause-setsminimal unsatisfiability

We consider a fundamental problem in the theory of branching heuristics for tree-based solvers, applicable e.g. to SAT, #SAT, CSP, #CSP. Such tree-based solvers are used as the cubing-part in the Cube-and-Conquer paradigm, and are thus of renewed interest for general (#)SAT solving. These solvers build at least implicitly a branching (backtracking) tree, with the goal to minimise tree-size. The heuristics are based on evaluating the progress made in a transition from an instance F to some “simplified” F′ by a distance d(F,F′) (the bigger the more progress). When a branching (F1′,⋯,Fk′) is to be chosen for F, for each possibility we consider its branching tuple t given by ti=d(F,Fi′), project it to a single number π(t), and choose a branching with minimal π(t). This paper investigates the choices for π(t), in a theoretical framework. The general theory is reviewed, together with the theoretical result on the “canonical projection” π(t)=τ(t). Focusing then on binary branchings (k=2, t=(a,b)), we analyse the asymptotics of τ(a,b), and reflect on the whole possible range of binary projections, arriving at first practical possibilities for dynamic heuristics.

We consider minimally unsatisfiable 2-CNFs, i.e., minimally unsatisfiable conjunctive normal forms, where each clause contains at most 2 literals (short 2-MUs). Characterisations of 2-MUs in the literature have been restricted to the nonsingular case (where every variable occurs positively and negatively at least twice), and those with a unit-clause. We provide the full classification of 2-MUs, and obtain polytime isomorphism decision. The main tool used here is the implication digraph, which allows to reduce the problem of determining the isomorphism types of 2-MUs to the isomorphism types of "weak double cycles" (WDCs), obtained from the digraph versions of undirected cycles by splitting vertices and arcs. We show that these digraphs have at most one skew-symmetry, and thus the skew-symmetric WDCs can be considered as 2-CNFs.

Autarkies for SAT can be used for theoretical studies, pre-processing and inprocessing. They generalise satisfying assignments by allowing to leave some clauses "untouched" (no variable assigned). We introduce the natural generalisation to DQCNF (dependency-quantified boolean CNF), with the perspective of SAT translations for special cases. Finding an autarky for DQCNF is as hard as finding a satisfying assignment. Fortunately there are (many) natural autarky-systems, which allow restricting the range of autarkies to a more feasible domain, while still maintaining the good general properties of arbitrary autarkies. We discuss what seems the most fundamental autarky systems, and how the related reductions can be found by SAT solvers.

This paper considers unsatisfiable CNF formulas and addresses the problem of computing the union of the clauses included in some minimally unsatisfiable subformula (MUS). The union of MUSes represents a useful notion in infeasibility analysis since it summarizes all the causes for the unsatisfiability of a given formula. The paper proposes a novel algorithm for this problem, developing a refined recursive enumeration of MUSes based on powerful pruning techniques. Experimental results indicate the practical suitability of the approach.