Hiro Ito

An improved constant-time approximation algorithm for maximum~matchings

This paper studies approximation algorithms for problems on degree-bounded graphs. Let n and d be the number of vertices and the degree bound, respectively. This paper presents an algorithm to approximate the size of some maximal independent set with additive error ε n whose running time is O(d2). Using this algorithm, it also shows that there are approximation algorithms for many other problems, e.g., the maximum matching problem, the minimum vertex cover problem, and the minimum set cover problem, that run exponentially faster than existing algorithms with respect to d and 1/ε. Its approximation algorithm for the maximum matching problem can be transformed to a testing algorithm for the property of having a perfect matching with two-sided error. On the contrary, it also shows that every one-sided error tester for the property requires at least Ω(n) queries.

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.

Edge connectivity between nodes and node‐subsets

Let G = (V, E) be a graph where V and E are a set of nodes and a set of edges, respectively. Let X = {V1, V2, …, Vp}, Vi ⊆ V be a family of node-subsets. Each node-subset Vi is called an area, and a pair of G and X is called an area graph. A node ν ∈ V and an area Vi ∈ X are called k-NA (node-to-area)-connected if the minimum size of a cut separating ν and Vi is at least k. We say that an area graph (G, X) is k-NA-edge-connected when each ν ∈ V and Vi ∈ X are k-NA-edge-connected. This paper gives a necessary and sufficient condition for a given (G, X) to be k-NA-edge-connected: (G, X) is k-NA-edge-connected iff, for all positive integers h ≤ k, every h-edge-connected component of G includes at least one node from each area or has at least k edges between the component and the rest of the nodes. This paper also studied the Minimum Area Augmentation Problem, i.e., the problem of determining whether or not a given area graph (G, X) is k-NA-edge-connected and of choosing the minimum number of nodes to be included in appropriate areas to make the area graph k-NA-edge-connected (if (G, X) is not k-NA-edge-connected). This problem can be regarded as one of the location problems, which arises from allocating service-nodes on multimedia networks. We propose an O(|E| + |V|2 + L′ + min {|E|, k|V|} min {k|V|, k + |V|2}) time algorithm for solving this problem, where L′ is a space required to represent output areas. For a fixed k, this algorithm also runs in linear time when the h-edge-connected components of G are available for all h = 1, 2, …, k.

Linear-time enumeration of isolated cliques

For a given graph G of n vertices and m edges, a clique S of size k is said to be c-isolated if there are at most ck outgoing edges from S. It is shown that this parameter c is an interesting measure which governs the complexity of finding cliques. In particular, if c is a constant, then we can enumerate all c-isolated maximal cliques in linear time, and if c = O(log n), then we can enumerate all c-isolated maximal cliques in polynomial time. Note that there is a graph which has a superlinear number of c-isolated cliques if c is not a constant, and there is a graph which has a superpolynomial number of c-isolated cliques if c = ω(log n). In this sense our algorithm is optimal for the linear-time and polynomial-time enumeration of c-isolated cliques.

Source location problems considering vertex-connectivity and edge-connectivity simultaneously

Let G = (V, E) be an undirected multigraph, where V and E are a set of vertices and a set of edges, respectively. Let k and l be fixed nonnegative integers. This paper considers location problems of finding a minimum-size vertex-subset S ⊆ V such that for each vertex x ∈ V the vertex-connectivity between S and x is greater than or equal to k and the edge-connectivity between S and x is greater than or equal to l. For the problem with edge-connectivity requirements, that is, k = 0, an O(L(|V|, |E|, l)) time algorithm is already known, where L(|V|, |E|, l) is the time to find all h-edge-connected components for h = 1, 2, … , l and O(L(|V|, |E|, l)) = O(|E| + |V|2 + |V|min{|E|, l|V|}min{l, |V|}) is known. In this paper, we show that the problem with k ≥ 3 is NP-hard even for l = 0. We then present an O(L(|V|, |E|, l)) time algorithm for 0 ≤ k ≤ 2 and l ≥ 0. Moreover, we prove that the problem parameterized by the size of S is fixed-parameter tractable (FPT) for k = 3 and l ≥ 0.

2-Dimension Ham Sandwich Theorem for Partitioning into Three Convex Pieces

Let m ≥ 2, n ≥ 2 and q ≥ 2 be positive integers. Let S r and S b be two disjoint sets of points in the plane such that no three points of S r ∪ S b are collinear, |S r | = nq, and |S b | = mq. This paper shows that Kaneko and Kano’s conjecture is true, i.e., S r ∪ S b can be partitioned into q subsets P 1,P 2,...,P q satisfying that: (i) conv(P i ) ∩ conv(P j ) = ∅ for all 1 ≤ i < j ≤ q; (ii) |P i ∩ S r |= n and |P i ∩ S b | = m for all 1 ≤ i ≤ q. This is a generalization of 2-dimension Ham Sandwich Theorem.

Enumeration of isolated cliques and pseudo-cliques

In this article, we consider <i>isolated</i> cliques and <i>isolated</i> dense subgraphs. For a given graph <i>G</i>, a vertex subset <i>S</i> of size <i>k</i> (and also its induced subgraph <i>G</i>(<i>S</i>)) is said to be <i>c</i>-isolated if <i>G</i>(<i>S</i>) is connected to its outside via less than <i>ck</i> edges. The number <i>c</i> is sometimes called the <i>isolation factor</i>. The subgraph appears more isolated if the isolation factor is smaller. The main result in this work shows that for a fixed constant <i>c</i>, we can enumerate all <i>c</i>-isolated maximal cliques (including a maximum one, if any) in linear time. In more detail, we show that, for a given graph <i>G</i> of <i>n</i> vertices and <i>m</i> edges, and a positive real number <i>c</i>, all <i>c</i>-isolated maximal cliques can be enumerated in time <i>O</i>( c⁴ 2^2c<i>m</i>). From this, we can see that: (1) if <i>c</i> is a constant, all <i>c</i>-isolated maximal cliques can be enumerated in linear time, and (2) if <i>c</i> &equlas; <i>O</i>(log <i>n</i>), all <i>c</i>-isolated maximal cliques can be enumerated in polynomial time. Moreover, we show that these bounds are tight. That is, if <i>f</i>(<i>n</i>) is an increasing function not bounded by any constant, then there is a graph of <i>n</i> vertices and <i>m</i> edges for which the number of <i>f</i>(<i>n</i>)-isolated maximal cliques is superlinear in <i>n</i> + <i>m</i>. Furthermore, if <i>f</i>(<i>n</i>) = ω(log <i>n</i>), there is a graph of <i>n</i> vertices and <i>m</i> edges for which the number of <i>f</i>(<i>n</i>)-isolated maximal cliques is superpolynomial in <i>n</i> + <i>m</i>. We next introduce the idea of pseudo-cliques. A <i>pseudo-clique</i> having an average degree α and a minimum degree β, denoted by <i>PC</i>(α,β), is a set <i>V</i>′ ⊆ <i>V</i> such that the subgraph induced by <i>V</i>′ has an average degree of at least α and a minimum degree of at least β. This article investigates these, and obtains some cases that can be solved in polynomial time and some other cases that have a superpolynomial number of solutions. Especially, we show the following results, where <i>k</i> is the number of vertices of the isolated pseudo-cliques: (1) For any ϵ > 0 there is a graph of <i>n</i> vertices for which the number of 1-isolated <i>PC</i>(<i>k</i> − (log <i>k</i>)^{1 + ϵ}, <i>k</i>/(log <i>k</i>)^{1 + ϵ}) is superpolynomial, and (2) there is a polynomial-time algorithm which enumerates all <i>c</i>-isolated <i>PC</i>(<i>k</i> − log <i>k</i>, <i>k</i>/log <i>k</i>), for any constant <i>c</i>.

Improved Constant-Time Approximation Algorithms for Maximum Matchings and Other Optimization Problems

We study constant-time approximation algorithms for bounded-degree graphs, which run in time independent of the number of vertices

Linear time algorithms for graph search and connectivity determination on complement graphs

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Property Testing on k -Vertex-Connectivity of Graphs

. We present an algorithm that decides whether a vertex is contained in a some fixed maximal independent set with expected query complexity