Toshimasa Ishii

Minimum cost source location problem with vertex-connectivity requirements in digraphs

Abstract Given a digraph (or an undirected graph) G=(V,E) with a set V of vertices v with nonnegative real costs w(v), and a set E of edges and a positive integer k, we deal with the problem of finding a minimum cost subset S⊆V such that, for each vertex v∈V−S, there are k vertex-disjoint paths from S to v. In this paper, we show that the problem can be solved by a greedy algorithm in O ( min {k, n }nm) time in a digraph (or in O ( min {k, n }kn 2 ) time in an undirected graph), where n=|V| and m=|E|. Based on this, given a digraph and two integers k and l, we also give a polynomial time algorithm for finding a minimum cost subset S⊆V such that for each vertex v∈V−S, there are k vertex-disjoint paths from S to v as well as l vertex-disjoint paths from v to S.

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G = (V, E), a family W of sets W ⊆ V of vertices (areas), and a requirement function r : W → Z+ (where Z+ is the set of nonnegative integers) we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v ∈ V and an area W ∈ W. So far this problem was shown to be NP-hard in the uniform case of r(W) = 1 for each W ∈ W, and polynomially solvable in the uniform case of r(W) = r ≥ 2 for each W ∈ W. In this paper, we show that the problem can be solved in O(m + pn4(r* + logn)) time, even if r(W) ≥ 2 holds for each W ∈ W, where n = |V|, m = |{{u, v}|(u, v) ∈ E}|, p = |W|, and r* = max{r(W)|W ∈ W}.

Augmenting forests to meet odd diameter requirements

Given a graph G=(V,E) and an integer D>=1, we consider the problem of augmenting G by the smallest number of new edges so that the diameter becomes at most D. It is known that no constant approximation algorithms to this problem with an arbitrary graph G can be obtained unless P=NP. For a forest G and an odd D>=3, it was open whether the problem is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with a forest G and an odd D; our algorithm delivers an 8-approximate solution in O(|V|^3) time. We also show that a 4-approximate solution to the problem with a forest G and an odd D can be obtained in linear time if the augmented graph is additionally required to be biconnected.

A linear time algorithm for L(2,1)-labeling of trees

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x) − f(y)| ≥ 2 if x and y are adjacent and |f(x) − f(y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.75,Δ1.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n = |V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

On the minimum local-vertex-connectivity augmentation in graphs

Given a graph G and target values r(u,v) prescribed for each pair of vertices u and v, we consider the problem of augmenting G by a smallest set F of new edges such that the resulting graph G+F has at least r(u,v) internally disjoint paths between each pair of vertices u and v. We show that the problem is NP-hard even if for some constant k ≥ 2 G is (k - 1 )-vertex-connected and r(u,v) ∈ {0,k} holds for u,v ∈ V. We then give a linear time algorithm which delivers a 3/2-approximation solution to the problem with a connected graph G and r(u, v) ∈ {0, 2}, u, v ∈ V.

Augmenting Outerplanar Graphs to Meet Diameter Requirements

Given a graph G = (V, E) and an integer D ≥ 1, we consider the problem of augmenting G by a minimum set of new edges so that the diameter becomes at most D. It is known that no constant factor approximation algorithms to this problem with an arbitrary graph G can be obtained unless P = NP, while the problem with only a few graph classes such as forests is approximable within a constant factor. In this paper, we give the first constant factor approximation algorithm to the problem with an outerplanar graph G. We also show that if the target diameter D is even, then the case where G is a partial 2-tree is also approximable within a constant.

On the Minimum Augmentation of an l-Connected Graph to a k-Connected Graph

Given an undirected graph G = (V, E) and a positive integer k, we consider the problem of augmenting G by the smallest number of new edges to obtain a k-vertex-connected graph. In this paper, we show that, for k ≥ 4 and k ≥ l+2, an l-vertex-connected graph G can be made k-vertex-connected by adding at most δ(k-1)+max{0, (δ-1)(l-3)-1} surplus edges over the optimum in O(δ(k2n2 + k3n3/2)) time, where δ = k - l and n = |V|.

Greedy approximation for the source location problem with vertex-connectivity requirements in undirected graphs

Let G = ( V , E ) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v � V has a demand d ( v ) � Z + , and a cost c ( v ) � R + , where Z + and R + denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing � v � S c ( v ) such that there are at least d ( v ) pairwise vertex-disjoint paths from S to v for each vertex v � V - S . It is known that the problem is not approximable within a ratio of O ( ln � v � V d ( v ) ) , unless NP has an O ( N log log N ) -time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and d � = 4 holds, then the problem is NP-hard, where d � = max { d ( v ) | v � V } .In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a max { d � , 2 d � - 6 } -approximate solution to the problem in O ( min { d � , | V | } d � | V | 2 ) time, while we also show that there exists an instance for which it provides no better than a ( d � - 1 ) -approximate solution. Especially, in the case of d � ≤ 4 , we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to d � ≤ 4 .

The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, ..., k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2, 1)-total labeling number λ2T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2T(G) ≤ Δ(G)+2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2T(G) ≤ Δ(G)+2 even in the case of Δ(G) ≤ 4.

The source location problem with local 3-vertex-connectivity requirements

Let G=(V,E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v@?V has an integer valued demand d(v)>=0. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices with the minimum cardinality such that there are at least d(v) vertex-disjoint paths between S and each vertex v@?V-S. In this paper, we show that the problem with d(v)= =4 for some vertex v@?V, the problem is NP-hard.