The search for natural NP-intermediate problems is one of the holy grails within computational complexity. Ladners original diagonalization technique for generating NP-intermediate problems, blowing holes, has a serious shortcoming: it creates problems with a highly artificial structure by arbitrarily removing certain problem instances. In this article we limit this problem by generalizing Ladners method to use parameters with various characteristics. This allows one to define more fine-grained parameters, resulting in NP-intermediate problems where we only blow holes in a controlled subset of the problem. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute (such as tree-width) are covered, thereby generalizing a result by Grohe. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Gamma such that CSP(Gamma) are NP-intermediate. The sets Gamma can have very different properties compared to previous constructions (by, for instance, Bodirsky and Grohe) and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems. This settles an open question posed by Nordh and Zanuttini. (C) 2015 Elsevier B.V. All rights reserved.

We consider methods for constructing NP-intermediate problems under the assumption that P ≠ NP. We generalize Ladner’s original method for obtaining NP-intermediate problems by using parameters with various characteristics. In particular, this generalization allows us to obtain new insights concerning the complexity of CSP problems. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute (such as tree-width) are covered. Hereby we generalize a result by Grohe on width parameters and NP-intermediate problems. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Γ such that CSP(Γ) are NP-intermediate. The sets Γ can have very different properties compared to previous constructions (by, for instance, Bodirsky & Grohe) and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems. This settles an open question posed by Nordh & Zanuttini.

We study the complexity of the propositional minimal inference problem. Although the complexity of this problem has been already extensively studied before because of its fundamental importance in nonmonotonic logics and commonsense reasoning, no complete classification of its complexity was found. We classify the complexity of four different and well-studied formalizations of the problem in the version with unbounded queries, proving that the complexity of the minimal inference problem for each of them has a trichotomy (between P, coNP-complete, and I P-2-complete). One of these results finally settles with a positive answer the trichotomy conjecture of Kirousis and Kolaitis (Theory Comput. Syst. 37(6):659-715, 2004). In the process we also strengthen and give a much simplified proof of the main result from Durand and Hermann (Proceedings 20th Symposium on Theoretical Aspects of Computer Science (STACS 2003), pp. 451-462, 2003).

The purpose of this note is to give upper bounds (assuming P different from NP) on how far the generalizations of Skolem sequences can be taken while still hoping to resolve the existence question. We prove that the existence questions for both multi-Skolem sequences and generalized Skolem sequences are strongly NP-complete. These results are significant strengthenings and simplifications of the recent NP-completeness result for generalized multi-Skolem sequences.

We study a family of problems, called Maximum Solution (Max Sol), where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the complexity and approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering the Max Sol problem for relations defined by regular signed logic over finite subsets of the natural numbers; the complexity of the corresponding decision problem has recently been classified by Creignou et al. (Theory Comput. Syst. 42(2):239–255, 2008). We give sufficient conditions for when such problems are polynomial-time solvable and we prove that they are APX-hard otherwise. Similar dichotomies are also obtained for variants of the Max Sol problem.

For a fixed graph H, let RET(H) denote the problem of deciding whether a given input graph is retractable to H. We classify the complexity of RET(H) when H is a graph (with loops allowed) where each connected component has at most one cycle, i.e., a pseudoforest. In particular, this result extends the known complexity classifications of RET(H) for reflexive and irreflexive cycles to general cycles. Our approach is based mainly on algebraic techniques from universal algebra that previously have been used for analyzing the complexity of constraint satisfaction problems.

We introduce and investigate the concept of frozen partial co-clones. Our main motivation for studying frozen partial co-clones is that they, have important applications in complexity analysis of constraints. The frozen partial co-clones lie between the co-clones and partial co-clones in the sense that the partial co-clone lattice is a refinement of the frozen partial co-clone lattice, which in turn is a refinement of the co-clone lattice. We concentrate on the Boolean domain. and determine large parts of the frozen partial co-clone lattice.

We present an efficient algorithm to find an optimal integer solution of a given system of 2-variable equalities and I-variable inequalities with respect to a given linear objective function. Our algorithm has worst-case running time in O(N-2) where N is the number of bits in the input.