In this paper, we study a variant of the MINIMUM DOMINATING SET problem. Given an unweighted undirected graph G=(V,E) of n=|V| vertices, the goal of the MINIMUM SINGLE DOMINATING CYCLE problem (MinSDC) is to find a single shortest cycle which dominates all vertices, i.e., a cycle C such that for the set V(C) of vertices in C and the set N(V(C)) of neighbor vertices of C, V(G)=V(C)∪N(V(C)) and |V(C)| is minimum over all dominating cycles in G [6], [17], [24]. In this paper we consider the (in)approximability of MinSDC if input graphs are restricted to some special classes of graphs. We first show that MinSDC is still NP-hard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r≥3). Then, we show the (lnn+1)-approximability and the (1-ε)lnn-inapproximability of MinSDC on split graphs under P≠NP. Furthermore, we explicitly design a linear-time algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running time-bound.

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.

This paper introduces a new timetabling problem on universities, called interview timetabling. In this problem, some constant number, say three, of referees are assigned to each of 2n graduate students. Our task is to construct a presentation timetable of these 2n students using n timeslots and two rooms, so that two students evaluated by the same referee must be assigned to different timeslots. The optimization goal is to minimize the total number of movements of all referees between two rooms. This problem comes from the real world in the interview timetabling in Kyoto University. We propose two restricted models of this problem, and investigate their time complexities.

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 two dimensional mesh is widely considered to be a promising parallel architecture in its scalability. In this architecture, processors are naturally placed at intersections of horizontal and vertical grids, while there can be three different types of communication links: (i) The first type is the most popular model, called a mesh-connected computer: Each processor is connected to its four neighbours by local connections. (ii) Each processor of the second type is connected to a couple of (row and column) buses. The system is then called a mesh of buses. (iii) The third model is equipped with both buses and local connections, which is called a mesh-connected computer with buses. Mesh routing has received considerable attention for the last two decades, and a variety of algorithms have been proposed. This paper provides an overview of lower and upper bounds for algorithms, with pointers to the literature, and suggests further research directions for mesh routing.

In this paper, we investigate the maximum induced matching problem (MaxIM) on C5-free d-regular graphs. The previously known best approximation ratio for MaxIM on C5-free d-regular graphs is $left(rac{3d}{4}-rac{1}{8}+rac{3}{16d-8} ight)$. In this paper, we design a $left(rac{2d}{3}+rac{1}{3} ight)$-approximation algorithm, whose approximation ratio is strictly smaller/better than the previous one when d≥6.

This paper studies generalized variants of the MAXIMUM INDEPENDENT SET problem, called the MAXIMUM DISTANCE-d INDEPENDENT SET problem (MaxDdIS for short). For an integer d≥2, a distance-d independent set of an unweighted graph G=(V, E) is a subset S⊆V of vertices such that for any pair of vertices u, v∈S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r≥3) and planar graphs, as follows: (1) For every fixed integers d≥3 and r≥3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d=r=3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

In this paper we discuss approximation algorithms for the ELEMENT-DISJOINT STEINER TREE PACKING problem (Element-STP for short). For a graph G=(V,E) and a subset of nodes T⊆V, called terminal nodes, a Steiner tree is a connected, acyclic subgraph that contains all the terminal nodes in T. The goal of Element-STP is to find as many element-disjoint Steiner trees as possible. Element-STP is known to be APX-hard even for |T|=3 [1]. It is also known that Element-STP is NP-hard to approximate within a factor of Ω(log |V|) [3] and there is an O(log |V|)-approximation algorithm for Element-STP [2],[4]. In this paper, we provide a $lceil rac{|T|}{2} ceil$-approximation algorithm for Element-STP on graphs with |T| terminal nodes. Furthermore, we show that the approximation ratio of 3 for Element-STP on graphs with five terminal nodes can be improved to 2.

In difficult classification problems of the z-dimensional points into two groups giving 0-1 responses due to the messy data structure, we try to find the denser regions for the favorable customers of response 1, instead of finding the boundaries to separate the two groups. Such regions are called the bumps, and finding the boundaries of the bumps is called the bump hunting. The main objective of this paper is to find the largest region of the bumps under a specified ratio of the number of the points of response 1 to the total. Then, we may obtain a trade-off curve between the number of points of response 1 and the specified ratio. The decision tree method with the Gini's index will provide the simple-shaped boundaries for the bumps if the marginal density for response 1 shows a rather simple or monotonic shape. Since the computing time searching for the optimal trees will cost much because of the NP-hardness of the problem, some random search methods, e.g., the genetic algorithm adapted to the tree, are useful. Due to the existence of many local maxima unlike the ordinary genetic algorithm search results, the extreme-value statistics will be useful to estimate the global optimum number of captured points; this also guarantees the accuracy of the semi-optimal solution with the simple descriptive rules. This combined method of genetic algorithm search and extreme-value statistics use is new. We apply this method to some artificial messy data case which mimics the real customer database, showing a successful result. The reliability of the solution is discussed.