Yuichi Asahiro

Greedily Finding a Dense Subgraph

Given an n-vertex graph with non-negative edge weights and a positive integer k ≤ n, we are to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2+n/(2k))2-O(1/n) ≤ R ≤ (1/2+n/(2k))2+O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k2) for k<n/3. For k = n/2, for example, these bounds are 9/4+=O(1/n), improving on naive lower and upper bounds of 2 and 4 respectively. The upper bound for general k shows that this simple algorithm is better than the best previously known algorithm at least by a factor of 2 when k ≥ n11/18.

Approximating maximum diameter-bounded subgraphs

The paper studies the maximum diameter-bounded subgraph problem (MaxDBS for short) which is defined as follows: Given an n-vertex graph G and a fixed integer d≥1, the goal is to find its largest subgraph of the diameter d. If d=1, the problem is identical to the maximum clique problem and thus it is

NP-hardness of the sorting buffer problem on the uniform metric

{\cal NP}

Independentand cooperative parallel search methods for the generalized assignment problem

-hard to approximate MaxDBS to within a factor of n1−e for any e>0. Also, it is known to be

A Self-stabilizing Marching Algorithm for a Group of Oblivious Robots

{\cal NP}

A distributed ladder transportation algorithm for two robots in a corridor

-hard to approximate MaxDBS to within a factor of n1/3−e for any e>0 and a fixed d≥2. In this paper, we first strengthen the hardness result; we prove that, for any e>0 and a fixed d≥2, it is

Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation

{\cal NP}

Approximation Algorithms for the Graph Orientation Minimizing the Maximum Weighted Outdegree

-hard to approximate MaxDBS to within a factor of n1/2−e. Then, we show that a simple polynomial-time algorithm achieves an approximation ratio of n1/2 for any even d≥2, and an approximation ratio of n2/3 for any odd d≥3. Furthermore, we investigate the (in)tractability and the (in)approximability of MaxDBS on subclasses of graphs, including chordal graphs, split graphs, interval graphs, and k-partite graphs.

Pickup and delivery for moving objects on broken lines

An instance of the sorting buffer problem (SBP) consists of a sequence of requests for service, each of which is specified by a point in a metric space, and a sorting buffer which can store up to a limited number of requests and rearrange them. To serve a request, the server needs to visit the point where serving a request p following the service to a request q requires the cost corresponding to the distance d(p,q) between p and q. The objective of SBP is to serve all input requests in a way that minimizes the total distance traveled by the server by reordering the input sequence. In this paper, we focus our attention to the uniform metric, i.e., the distance d(p,q)=1 if p q, d(p,q)=0 otherwise, and present the first NP-hardness proof for SBP on the uniform metric.

Finding Dense Subgraphs

The generalized assignment problem is a representative NP-hard problem, for which many heuristic algorithms are known. In this article, two parallel heuristic algorithms are proposed, which are based on the ejection chain local search (EC) proposed by Yagiura et al. One is a simple parallelization called multistart parallel EC (MPEC) and the other is cooperative parallel EC (CPEC). In MPEC each search process independently explores search space while in CPEC search processes share partial information to cooperate with each other. The experimental results with 9 computers for large benchmark instances show that (1) MPEC and CPEC, respectively, run twice and 4 times faster than EC, and (2) compared to EC, the difference in quality between obtained solutions and theoretical lower bounds is reduced to [Formula: See Text] and [Formula: See Text] by MPEC and CPEC, respectively. It is said that these methods give us full benefit of parallelization, speedup and improvement for quality of solutions.