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Abstract: Upward-closed sets of integer vectors enjoy the merit of having a finite number of minimal elements, which is behind the decidability of a number of Petri net related problems. In general, however, such a finite set of minimal elements may not be effectively computable. In this paper, we develop a unified strategy for computing the sizes of the minimal elements of certain upward-closed sets associated with Petri nets. Our approach can be regarded as a refinement of a previous work by Valk and Jantzen (in which a necessary and sufficient condition for effective computability of the set was given), in the sense that complexity bounds now become available provided that a bound can be placed on the size of a witness for a key query. The sizes of several upward-closed sets that arise in the theory of Petri nets as well as in backward-reachability analysis in automated verification are derived in this paper, improving upon previous decidability results shown in the literature. [Copyright &y& Elsevier]

Abstract: We investigate the problem of characterizing the solution spaces for timed automata augmented by unknown timing parameters (called timing parameter automata (TPA)). The main contribution of this paper is that we identify three non-trivial subclasses of TPAs, namely, upper-bound, lower-bound and bipartite TPAs, and analyze how hard it is to characterize the solution spaces. As it turns out, we are able to give complexity bounds for the sizes of the minimal (resp., maximal) elements which completely characterize the upward-closed (resp., downward-closed) solution spaces of upper-bound (resp., lower-bound) TPAs. For bipartite TPAs, it is shown that their solution spaces are not semilinear in general. We also extend our analysis to TPAs equipped with counters without zero-test capabilities. [Copyright &y& Elsevier]

Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends in O and covers at most k points from P. The k-tour cover problem is known to be $\mathcal{N}\mathcal{P}$-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomial-time approximation scheme (PTAS) for small values of k, k = Ø(log n/log log n). In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k ≤ 2logδ n, where δ = δ(ε). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with Ø((k/ε)Ø(1)) points. [ABSTRACT FROM AUTHOR]

We study the problem of finding all maximal approximate gapped palindromes in a string. More specifically, given a string S of length n, a parameter q ≥ 0 and a threshold k > 0, the problem is to identify all substrings in S of the form uvw such that (1) the Levenshtein distance between ur and w is at most k, where ur is the reverse of u and (2) v is a string of length q. The best previous work requires O(k2n) time. In this paper, we propose an O(kn)-time algorithm for this problem by utilizing an incremental string comparison technique. It turns out that the core technique actually solves a more general incremental string comparison problem that allows the insertion, deletion, and substitution of multiple symbols. [ABSTRACT FROM AUTHOR]