This paper introduces the maximization version of the k-path vertex cover problem, called the MAXIMUM K-PATH VERTEX COVER problem (MaxPkVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path Pk includes a vertex v in a vertex set S, then we say that v or S covers Pk. Given a graph G=(V, E) and an integer s, the goal of MaxPkVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxPkVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxPkVC on subclasses of graphs. We prove that MaxP3VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP3VC can be solved in polynomial time.

The family of stable matching problems have been well-studied across a wide field of research areas, including economics, mathematics and computer science. In general, an instance of a stable matching problem is given by a set of participants who have expressed their preferences of each other, and asks to find a “stable” matching, that is, a pairing of the participants such that no unpaired participants prefer each other to their assigned partners. In the case of the Stable Roommates Problem (SR), it is known that given an even number n of participants, there might not exist a stable matching that pairs all of the participants, but there exist efficient algorithms to determine if this is possible or not, and if it is possible, produce such a matching. Common extensions of SR allow for the participants' preference lists to be incomplete, or include indifference. Allowing indifference in turn, gives rise to different possible definitions of stability, super, strong, and weak stability. While instances asking for super and strongly stable matching can be efficiently solved even if preference lists are incomplete, the case of weak stability is NP-complete. We examine a restricted case of indifference, introducing the concept of unranked entries. For this type of instances, we show that the problem of finding a weakly stable matching remains NP-complete even if each participant has a complete preference list with at most two unranked entries, or is herself unranked for up to three other participants. On the other hand, for instances where there are m acceptable pairs and there are in total k unranked entries in all of the participants' preference lists, we propose an O(2kn2)-time and polynomial space algorithm that finds a stable matching, or determines that none exists in the given instance.

Users of wireless mobile devices need Internet access not only when they stay at home or office, but also when they travel. It may be desirable for such users to select a "longcut route" from their current location to his/her destination that has longer travel time than the shortest route, but provides a better mobile wireless environment. In this paper, we formulate the above situation as the optimization problem of “optimal longcut route selection”, which requires us to find the best route concerning the wireless environment subject to a travel time constraint. For this new problem, we show NP-hardness, propose two pseudo-polynomial time algorithms, and experimental evaluation of the algorithms.

We prove NP-completeness of deciding whether a given loop of colored right isosceles triangles, hinged together at edges, can be folded into a specified rectangular three-color pattern. By contrast, the same problem becomes polynomially solvable with one color or when the target shape is a tree-shaped polyomino.

Given a connected graph G = (V, E) on n vertices, the MAXIMUM r-REGULAR INDUCED CONNECTED SUBGRAPH (r-MaxRICS) problem asks for a maximum sized subset of vertices S ⊆ V such that the induced subgraph G[S] on S is connected and r-regular. It is known that 2-MaxRICS and 3-MaxRICS are NP-hard. Moreover, 2-MaxRICS cannot be approximated within a factor of n1-ε in polynomial time for any ε > 0 unless P= NP. In this paper, we show that r-MaxRICS are NP-hard for any fixed integer r ≥ 4. Furthermore, we show that for any fixed integer r ≥ 3, r-MaxRICS cannot be approximated within a factor of n1/6-ε in polynomial time for any ε > 0 unless P= NP.

In this paper, we study the minimum spanning tree problem with label selection, that is, the problem of finding a minimum spanning tree of a vertex-labeled graph where the weight of each edge may vary depending on the selection of labels of vertices at both ends. The problem is especially important as the application to mathematical OCR. It is shown that the problem is NP-hard. However, for the application to mathematical OCR, it is sufficient to deal with only graphs with small tree-width. In this paper, a linear-time algorithm for series-parallel graphs is presented. Since the minimum spanning tree problem with label selection is closely related to the generalized minimum spanning tree problem, their relation is discussed.

Petri nets with inhibitor arcs are referred to as inhibitor-arc Petri nets. It is shown that modeling capability of inhibitor-arc Petri nets is equivalent to that of Turing machines. The subject of this paper is the legal firing sequence problem (INLFS) for inhibitor-arc Petri nets: given an inhibitor-arc Petri net IN, an initial marking M0 and a firing count vector X, find a firing sequence δ such that its firing starts from M0 and each transition t appears in δ exactly X(t) times as prescribed by X. The paper is the first step of research for time complexity analysis and designing algorithms of INLFS, one of the most fundamental problems for inhibitor-arc Petri nets having more modeling capability than ordinary Peri nets. The recognition version of INLFS, denoted as RINLFS, means a decision problem, asking a "yes" or "no" answer on the existence of a solution δ to INLFS. The main results are the following (1) and (2). (1) Proving (1-1) and (1-2) when the underlying Petri net of IN is an unweighted state machine: (1-1) INLFS can be solved in pseudo-polynomial (O(|X|)) time for IN of non-adjacent type having only one special place called a rivet; (1-2) RINLFS is NP-hard for IN with at least three rivets; (2) Proving that RINLFS for IN whose underlying Petri net is unweighted and forward conflict-free is NP-hard. Heuristic algorithms for solving INLFS are going to be proposed in separate papers.

Semistructured data comprises irregular structure and has no a-priori database schema, therefore we encounter several problems such as inefficient data retrieval and wasteful data storage. To cope with such problems, some schema extraction algorithms over semistructured data have been proposed, in which data is modeled as an unordered tree. However, the order of elements is indispensable for document data, therefore we consider extracting an optimal database schema over an ordered tree. We consider an optimization problem to extract a smallest database schema such that the density of each class is no less than a given threshold, where the density of a class represents a similarity between the type of the class and those of the objects in the class. We first prove that the corresponding decision problem is strongly NP-complete, and show that another version of the problem is strongly NP-hard and belongs to Δ2 P. Then we show that for any r < 3/2, there is no polynomial-time r-approximation algorithm that solves the optimization problem unless P = NP. Finally, we propose a kind of class called bounded class that can be constructed efficiently, then show a polynomial-time algorithm for constructing a database schema by using bounded classes.

The subject of the paper is to give an overview and latest results on the Legal Firing Sequence Problem of Petri nets (LFS for short). LFS is very fundamental in the sense that it appears as a subproblem or a simpler form of various basic problems in Petri net theory, such as the well-known marking reachability problem, the minimum initial resource allocation problem, the liveness (of level 4) problem, the scheduling problem and so on. However, solving LFS generally is not easy: it is NP -hard even for Petri nets having very simple structures. This intractability of LFS may have been preventing us from producing efficient algorithms for those problems. So research on LFS from computational complexity point of view seems to be rewarding.

We consider single machine problems involving both specific and generalized due dates simultaneously. We show that a polynomial time algorithm exists for the problem of minimizing max {Lmax, LHmax}, where Lmax and LHmax are the maximum lateness induced by specific and generalized due dates, respectively. We also show that a simple efficient algorithm exists for the problem of minimizing the maximum lateness induced by due dates which are natural generalization of both specific and generalized due dates. In addition to the algorithmic results above, we show that the problems of minimizing max {LHmax, -Lmin} and max{Lmax, -LHmin} are NP-hard in the strong sense, where Lmin and LHmin are the minimum lateness induced by specific and generalized due dates, respectively.

The subject of the paper is the minimum initial marking problem for scheduling in timed Petri net PN: given a vector X of nonnegative integers, a P-invariant Y of PN and a nonnegative integer π, find an initial marking M minimizing the value YtrM among those initial marking M such that there is a scheduling σ having the total completion time τ(σ)π with respect M , X and PN (a sequence of transitions, with the first transition firable on M , such that every transition t can fire prescribed number X(t) of times). The paper shows NP-hardness of the problem and proposes two approximation algorithms with their experimental evaluation.