Abstract. Our motivation for this work is the remarkable discovery that many large-scale real-world graphs ranging from Internet and World Wide Web to social and biological networks exhibit a power-law distribution: the number of nodes yi of a given degree i is proportional to i −β where β > 0 is a constant that depends on the application domain. There is practical evidence that combinatorial optimization in power-law graphs is easier than in general graphs, prompting the basic theoretical question: Is combinatorial optimization in power-law graphs easy? Does the answer depend on the power-law exponent β? Our main result is the proof that many classical NP-hard graph-theoretic optimization problems remain NP-hard on power law graphs for certain values of β. In particular, we show that some classical problems, such as CLIQUE and COLORING, remains NP-hard for all β ≥ 1. Moreover, we show that all the problems that satisfy the so-called "optimal substructure property" remains NP-hard for all β > 0. This includes classical problems such as MIN VERTEX-COVER, MAX INDEPENDENT-SET, and MIN DOMINATING-SET. Our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems.

Our motivation for this work is the remarkable discovery that many large-scale real-world graphs ranging from Internet and World Wide Web to social and biological networks appear to exhibit a power-law distribution: the number of nodes y i of a given degree i is proportional to i −β where β > 0 is a constant that depends on the application domain. There is practical evidence that combinatorial optimization in power-law graphs is easier than in general graphs, prompting the basic theoretical question: Is combinatorial optimization in power-law graphs easy? Does the answer depend on the power-law exponent β? Our main result is the proof that many classical NP-hard graph-theoretic optimization problems remain NP-hard on power-law graphs for certain values of β. In particular, we show that some classical problems, such as CLIQUE and COLORING, remain NP-hard for all β ≥ 1. Moreover, we show that all the problems that satisfy the so-called "optimal substructure property" remain NP-hard for all β > 0. This includes classical problems such as MINIMUM VERTEX COVER, MAXIMUM INDEPENDENT SET, and MINIMUM DOMINATING SET. Our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems.

The use of the universal and existential quantifiers with the capability to express the concept of at least k or all but k, for a non-negative integer k, has been thoroughly studied in various kinds of logics. In classical logic there are counting quantifiers, in modal logics graded modalities, in description logics number restrictions.\ud Recently, the complexity issues related to the decidability of the μ-calculus, when the universal and existential quantifiers are augmented with graded modalities, have been investigated by Kupfermann, Sattler and Vardi. They have shown that this problem is ExpTime-complete.\ud In this paper we consider another extension of modal logic, the Computational Tree Logic CTL, augmented with graded modalities generalizing standard quantifiers and investigate the complexity issues, with respect to the model-checking problem. We consider a system model represented by a pointed Kripke structure and give an algorithm to solve the model-checking problem running in time O() which is hence tight for the problem (where |ϕ| is the number of temporal and boolean operators and does not include the values occurring in the graded modalities).\ud In this framework, the graded modalities express the ability to generate a user-defined number of counterexamples (or evidences) to a specification ϕ given in CTL. However these multiple counterexamples can partially overlap, that is they may share some behavior. We have hence investigated the case when all of them are completely disjoint. In this case we prove that the model-checking problem is both NP-hard and coNP-hard and give an algorithm for solving it running in polynomial space. We have thus studied a fragment of this graded-CTL logic, and have proved that the model-checking problem is solvable in polynomial time

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Abstract. The model checking problem for open systems (called module checking) has been intensively studied in the literature, both for finite-state and infinite-state systems. In this paper, we focus on pushdown module checking with respect to decidable fragments of the fully enriched µ-calculus. We recall that finite-state module checking with respect to fully enriched µ-calculus is undecidable and hence the extension of this problem to pushdown systems remains undecidable as well. On the contrary, for the fragments of the fully enriched µ-calculus we consider here, we show that pushdown module checking is decidable and solvable in double-exponential time in the size of the formula and in exponential time in the size of the system. This result is obtained by exploiting a classical automata-theoretic approach via pushdown nondeterministic parity tree automata. In particular, we reduce in exponential time our problem to the emptiness problem for these automata, which is known to be decidable in Exptime. As a key step of our algorithm, we show an exponential improvement of the construction of a nondeterministic parity tree automaton accepting all models of a formula of the considered logic. This result, does not only allow our algorithm to match the known lower bound, but also to investigate decision problems related to the fragments of the enriched µ-calculus in a greatly simplified manner.