On the Containment Problem for Linear Sets

It is well known that the containment problem (as well as the equivalence problem) for semilinear sets is log-complete at the second level of the polynomial hierarchy (where hardness even holds in dimension 1). It had been shown quite recently that already the containment problem for multi-dimensional linear sets is log-complete at the same level of the hierarchy (where hardness even holds when numbers are encoded in unary). In this paper, we show that already the containment problem for 1-dimensional linear sets (with binary encoding of the numerical input parameters) is log-hard (and therefore also log-complete) at this level. However, combining both restrictions (dimension 1 and unary encoding), the problem becomes solvable in polynomial time

The Complexity of Computing Minimal Unidirectional Covering Sets

Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure

Backdoors to Normality for Disjunctive Logic Programs

Over the last two decades, propositional satisfiability (SAT) has become one of the most successful and widely applied techniques for the solution of NP-complete problems. The aim of this paper is to investigate theoretically how Sat can be utilized for the efficient solution of problems that are harder than NP or co-NP. In particular, we consider the fundamental reasoning problems in propositional disjunctive answer set programming (ASP), Brave Reasoning and Skeptical Reasoning, which ask whether a given atom is contained in at least one or in all answer sets, respectively. Both problems are located at the second level of the Polynomial Hierarchy and thus assumed to be harder than NP or co-NP. One cannot transform these two reasoning problems into SAT in polynomial time, unless the Polynomial Hierarchy collapses. We show that certain structural aspects of disjunctive logic programs can be utilized to break through this complexity barrier, using new techniques from Parameterized Complexity. In particular, we exhibit transformations from Brave and Skeptical Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter of the instance and n the input size. In other words, the reduction is fixed-parameter tractable for parameter k. As the parameter k we take the size of a smallest backdoor with respect to the class of normal (i.e., disjunction-free) programs. Such a backdoor is a set of atoms that when deleted makes the program normal. In consequence, the combinatorial explosion, which is expected when transforming a problem from the second level of the Polynomial Hierarchy to the first level, can now be confined to the parameter k, while the running time of the reduction is polynomial in the input size n, where the order of the polynomial is independent of k.Comment: A short version will appear in the Proceedings of the Proceedings of the 27th AAAI Conference on Artificial Intelligence (AAAI'13). A preliminary version of the paper was presented on the workshop Answer Set Programming and Other Computing Paradigms (ASPOCP 2012), 5th International Workshop, September 4, 2012, Budapest, Hungar

Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the $\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula $\Phi(x)$, it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure

Observations on complete sets between linear time and polynomial time

AbstractThere is a single set that is complete for a variety of nondeterministic time complexity classes with respect to related versions of m-reducibility. This observation immediately leads to transfer results for determinism versus nondeterminism solutions. Also, an upward transfer of collapses of certain oracle hierarchies, built analogously to the polynomial-time or the linear-time hierarchies, can be shown by means of uniformly constructed sets that are complete for related levels of all these hierarchies. A similar result holds for difference hierarchies over nondeterministic complexity classes. Finally, we give an oracle set relative to which the nondeterministic classes coincide with the deterministic ones, for several sets of time bounds, and we prove that the strictness of the tape-number hierarchy for deterministic linear-time Turing machines does not relativize

Structural Average Case Complexity

AbstractLevin introduced an average-case complexity measure, based on a notion of “polynomial on average,” and defined “average-case polynomial-time many-one reducibility” among randomized decision problems. We generalize his notions of average-case complexity classes, Random-NP and Average-P. Ben-Davidet al. use the notation of 〈C, F〉 to denote the set of randomized decision problems (L, μ) such thatLis a set in C andμis a probability density function in F. This paper introduces Aver〈C, F〉 as the class of randomized decision problems (L, μ) such thatLis computed by a type-C machine onμ-average andμis a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= 〈NP, P-comp〉 and Average-P=Aver〈P, ∗〉, where P-comp denotes the set of density functions whose distributions are computable in polynomial time, and ∗ denotes the set of all density functions. Mainly studied are polynomial-time reductions between randomized decision problems: many–one, deterministic Turing and nondeterministic Turing reductions and the average-case versions of them. Based on these reducibilities, structural properties of average-case complexity classes are discussed. We give average-case analogues of concepts in worst-case complexity theory; in particular, the polynomial time hierarchy and Turing self-reducibility, and we show that all known complete sets for Random-NP are Turing self-reducible. A new notion of “real polynomial-time computations” is introduced based on average polynomial-time computations for arbitrary distributions from a fixed set, and it is used to characterize the worst-case complexity classesΔpkandΣpkof the polynomial-time hierarchy

On the Complexity of Pointer Arithmetic in Separation Logic

We investigate the complexity consequences of adding pointer arithmetic to separation logic. Specifically, we study an extension of the points-to fragment of symbolic-heap separation logic with sets of simple “difference constraints” of the form where x and y are pointer variables and k is an integer offset. This extension can be considered a practically minimal language for separation logic with pointer arithmetic. Most significantly, we find that, even for this minimal language, polynomial-time decidability is already impossible: satisfiability becomes -complete, while quantifier-free entailment becomes -complete and quantified entailment becomes -complete (where is the second class in the polynomial-time hierarchy). However, the language does satisfy the small model property, meaning that any satisfiable formula has a model, and any invalid entailment has a countermodel, of polynomial size, whereas this property fails when richer forms of arithmetical constraints are permitted