ASP(Q) extends Answer Set Programming (ASP) with Quantifiers over answer sets. In this paper, we focus on the class of ASP(Q) programs with two quantifiers and weak constraints, denoted as 2-ASP$^w$(Q). 2-ASP$^w$(Q) is a practically relevant fragment of ASP(Q) that is expressive enough to capture optimization problems up to the class $\Delta ^P_3$. On the theoretical side, we provide a complete complexity characterization of the main computational tasks for 2-ASP$^w$(Q) programs, including tight completeness results and the analysis of nontrivial cases that have not been addressed in previous works. On the practical side, we introduce novel strategies for computing (optimal) quantified answer sets in the CASPER system, that rely on a Counterexample-Guided Abstraction Refinement (CEGAR) technique tailored to ASP(Q). An experimental evaluation on hard benchmarks from different application domains shows that the proposed techniques are effective in practice.

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientationsof an undirected graph. For example, acyclic orientations with certain diameter constraints areclosely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientationswith path constraints, and discuss its use in the solution of the vertex coloring problem andsome versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-definingand the introduction of new classes of valid inequalities.