Complexity of min-max subsequence problems

We determine the computational complexity of the problem of ordering a set of n numbers, either into a sequence or a cycle, such that the maximum sum of any k successive numbers is minimal. Both problems are easy for k=2 and strongly NP-hard for any k?3. However, the two problems allow a polynomial-time approximation scheme that is linear in n.     

Perfect correspondences between dot-depth and polynomial-time hierarchies

We introduce the polynomial-time tree reducibility (ptt-reducibility) for formal languages. Our main result establishes a one–one correspondence between this reducibility and inclusions between complexity classes. More precisely, for languages B and C it holds that B ptt-reduces to C if and only if the unbalanced leaf-language class of B is robustly contained in the unbalanced leaf-language class of C. Formerly, such correspondence was only known for balanced leaf-language classes. Moreover, we show that restricted to regular languages, the levels 0, 1/2, 1, and 3/2 of the dot-depth hierarchy are closed under ptt-reducibility. Our results also have applications in complexity theory: We obtain the first gap theorem of leaf-language definability above the Boolean closure of NP.     

On the computational complexity of qualitative coalitional games

We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices, with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games (qcgs) are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining qcgs, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a qcg is non-empty is D^p₁-complete. (As an aside, we present what we believe is the first “natural” problem that is proven to be complete for D^p₂.) We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research.     

Complexity results for answer set programming with bounded predicate arities and implications

Answer set programming is a declarative programming paradigm rooted in logic programming and non-monotonic reasoning. This formalism has become a host for expressing knowledge representation problems, which reinforces the interest in efficient methods for computing answer sets of a logic program. The complexity of various reasoning tasks for general answer set programming has been amply studied and is understood quite well. In this paper, we present a language fragment in which the arities of predicates are bounded by a constant. Subsequently, we analyze the complexity of various reasoning tasks and computational problems for this fragment, comprising answer set existence, brave and cautious reasoning, and strong equivalence. Generally speaking, it turns out that the complexity drops significantly with respect to the full non-ground language, but is still harder than for the respective ground or propositional languages. These results have several implications, most importantly for solver implementations: Virtually all currently available solvers have exponential (in the size of the input) space requirements even for programs with bounded predicate arities, while our results indicate that for those programs polynomial space should be sufficient. This can be seen as a manifestation of the “grounding bottleneck” (meaning that programs are first instantiated and then solved) from which answer set programming solvers currently suffer. As a final contribution, we provide a sketch of a method that can avoid the exponential space requirement for programs with bounded predicate arities. Some results in this paper have been presented in preliminary form at KR 2004 [15].     

Machines that Can Output Empty Words

We propose the e-model for leaf languages which complements the known balanced and unbalanced concepts. Inspired by the neutral behavior of rejecting paths of NP machines, we allow transducers to output empty words. The paper explains several advantages of the new model. A central aspect is that it allows us to prove strong gap theorems: For any class that is definable in the e-model, either or . For the existing models, gap theorems, where they exist at all, only identify gaps for the definability by regular languages. We prove gaps for the general case, i.e., for the definability by arbitrary languages. We obtain such general gaps for NP, coNP, 1NP, and co1NP. For the regular case we prove further gap theorems for Σ₂^P, Π₂^P, and Δ₂^P. These are the first gap theorems for Δ₂^P. This work is related to former work by Bovet, Crescenzi, and Silvestri, Vereshchagin, Hertrampf et al., Burtschick and Vollmer, and Borchert et al. An extended abstract of this paper was presented at the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006).S. Travers supported by the Konrad-Adenauer-Stiftung.     

A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover-Delayer games

In this note we show that the asymmetric Prover-Delayer game developed in Beyersdorff et al. (2010) [2] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form ²Ω(nlogn) for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki (1999) [7] and Dantchev and Riis (2001) [5].     

On the computational complexity of coalitional resource games

We study Coalitional Resource Games (crgs), a variation of Qualitative Coalitional Games (qcgs) in which each agent is endowed with a set of resources, and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources. We investigate and classify the computational complexity of a number of natural decision problems for crgs, over and above those previously investigated for qcgs in general. For example, we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound (i.e., a limit value for every resource) is co-np-complete. We then investigate the relationship between crgs and qcgs, and in particular the extent to which it is possible to translate between the two models. We first characterise the complexity of determining equivalence between crgs and qcgs. We then show that it is always possible to translate any given crg into a succinct equivalent qcg, and that it is not always possible to translate a qcg into an equivalent crg; we establish some necessary and some sufficient conditions for a translation from qcgs to crgs to be possible, and show that even where an equivalent crg exists, it may have size exponential in the number of goals and agents of its source qcg.     

A metric for positional games

We define an extended real-valued metric, ρ, for positional games and prove that this class of games is a topological semigroup. We then show that two games are finitely separated iff they are path-connected and iff two closely related Conway games are equivalent. If two games are at a finite distance then this distance is bounded by the maximum difference of any two atoms found in the games. We may improve on this estimate when two games have the same form, as given by a form match. Finally, we show that if ρ(G,H)=∞ then for all X we have G+X H+X, a step towards proving cancellation for positional games.     

Complexity aspects of generalized Helly hypergraphs

In [V.I. Voloshin, On the upper chromatic number of a hypergraph, Australas. J. Combin. 11 (1995) 25-45], Voloshin proposed the following generalization of the Helly property. Let p?1, q?0 and s?0. A hypergraph H is (p,q)-intersecting when every partial hypergraph H^′⊆H formed by p or less hyperedges has intersection of cardinality at least q. A hypergraph H is (p,q,s)-Helly when every partial (p,q)-intersecting hypergraph H^′⊆H has intersection of cardinality at least s. In this work, we study the complexity of determining whether H is (p,q,s)-Helly.     

On the complexity of constrained Nash equilibria in graphical games

A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium. Although the existence of a Nash equilibrium is guaranteed in the mixed framework (i.e., when players select their actions in a randomized manner) in many real-world applications the existence of “any” equilibrium is not enough. Rather, it is often desirable to single out equilibria satisfying some additional requirements (in order, for instance, to guarantee a minimum payoff to certain players), which we call constrained Nash equilibria.In this paper, a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p. This setting is very useful for modeling large population games, where typically each player does not directly depend on all the players, and representing her utility function extensively is either inconvenient or infeasible.Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players’ interactions. The analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies, i.e., when players are forced to deterministically choose the action to perform. In particular, for this latter case, restrictions on players’ interactions and on constraints are identified, that make the computation of Nash equilibria an easy problem, for which polynomial and highly-parallelizable algorithms are presented.     

Hashiwokakero is NP-complete

In a Hashiwokakero puzzle, one must build bridges to connect a set of islands. We show that deciding the solvability of such puzzles is NP-complete.     

Complexity of DNA sequencing by hybridization

In the paper, the question of the complexity of the combinatorial part of the DNA sequencing by hybridization, is analyzed. Subproblems of the general problem, depending on the type of error (positive, negative), are distinguished. Since decision versions of the subproblems assuming only one type of error are trivial, complexities of the search counterparts are studied. Both search subproblems are proved to be strongly NP-hard, as well as their uniquely promised versions.     

Relations between Average-Case and Worst-Case Complexity

The consequences of the worst-case assumption NP=P are very well understood. On the other hand, we only know a few consequences of the analogous average-case assumption “NP is easy on average.” In this paper we establish several new results on the worst-case complexity of Arthur-Merlin games (the class AM) under the average-case complexity assumption “NP is easy on average.”

On the P versus NP intersected with co-NP question in communication complexity

We consider the analog of the P versus NP∩co-NP question for the classical two-party communication protocols where polynomial time is replaced by poly-logarithmic communication: if both a boolean function f and its negation ¬f have small (poly-logarithmic in the number of variables) nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition model of communication this question was answered by Aho, Ullman and Yannakakis in 1983: here P=NP∩co-NP.We show that in the best partition model of communication the situation is entirely different: here P is a proper subset even of RP∩co-RP. This, in particular, resolves an open question raised by Papadimitriou and Sipser in 1982.     

The Relative Complexity of Approximate Counting Problems

Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of efficient approximation algorithm known as an “FPRAS”, and (ii) those that are complete for #P with respect to approximation-preserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically defined subclass of #P.     

The Relative Complexity of Approximate Counting Problems

Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of efficient approximation algorithm known as an FPRAS, and (ii) those that are complete for #P with respect to approximation-preserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically defined subclass of #P.     

Equilibria problems on games: Complexity versus succinctness

We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others, the problems of deciding the existence of a pure Nash equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete, when the number of players is large and the number of actions for each player is constant, and that the problem is -complete when the number of players is a constant and the size of the action sets is exponential in the size of the game representation. Again when the game is described explicitly the problem is decidable in polynomial time.     

Complexity of homomorphisms to direct products of graphs

For a graph G, OALG asks whether or not an input graph H together with a partial map g:S→G, S⊆V(H), admits a homomorphism f:H→G such that f|_S=g. We show that for connected graphs G₁, G₂, OAL G₁×G₂ is in P if G₁ and G₂ are trees and NP-complete otherwise.     

Smaller superconcentrators of density 28

An N-superconcentrator is a directed, acyclic graph with N input nodes and N output nodes such that every subset of the inputs and every subset of the outputs of same cardinality can be connected by node-disjoint paths. It is known that linear-size and bounded-degree superconcentrators exist. Here it is proved that such superconcentrators exist (by a random construction of certain expander graphs as building blocks) having density 28 (where the density is the number of edges divided by N). The best known density before this paper was 34.2 [U. Schöning, Construction of expanders and superconcentrators using Kolmogorov complexity, J. Random Structures Algorithms 17 (2000) 64-77] or 33 [L.A. Bassalygo, Personal communication, 2004].