Transversals of subtree hypergraphs and the source location problem in digraphs

Abstract

A hypergraph H = (V, E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T . To find a minimum size transversal for a subtree hypergraph is, in general, NP-hard. In this paper, we show that if it is possible to decide if a set of vertices W ⊆ V is a transversal in time S(n) ( where n = |V | ), then it is possible to find a minimum size transversal in O(n 3 S(n)). This result provides a polynomial algorithm for the Source Location Problem : a set of (k, l)-sources for a digraph D = (V, A) is a subset K of V such that for any v ∈ V \ K there are k arc-disjoint paths that each join a vertex of K to v and l arc-disjoint paths that each join v to K. The Source Location Problem is to find a minimum size set of (k, l)-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case S(n) is polynomial.

Minimum Cost Source Location Problems with Flow Requirements

“…But, it remained open to prove the NP-hardness in the strong sense or to devise a pseudo-polynomial time algorithm. For directed networks, Ito et al [11] showed that the problem is strongly NP-hard, even if either the cost function c or the demand functions d − and d + are uniform, and Bárász et al [3] and Heuvel et al [20] provided a polynomial time algorithm if c, d − and d + are all uniform. Tables (a) and (b) in Fig.…”

Embeddings Between Hypercubes and Hypertrees

“…A hypertree is a hypergraph H if there is a tree T such that the hyperedges of H induce subtrees in T [5]. In the literature, hypertree is also called a subtree hypergraph or arboreal hypergraph [5,23].…”

Source Location Problems with Flow Requirements

“…On the other hand, Sakashita et al [14] showed that the problem is in general NP-hard. For directed networks, Ito et al [10] showed that the problem is strongly NP-hard, even if either the cost function c or the demand functions d − and d + are uniform, and Bárász et al [3] and Heuvel et al [5] provided a polynomial time algorithm if c, d − and d + are all uniform. Tables (a) and (b) in Figure 1 summarize the best known bounds for the source location problems with the arc-connectivity requirements.…”

“…Location problems in networks are often formulated as optimization problems to determine the best location of facilities such as industrial plants or warehouses in given networks to satisfy a certain property. Location problems based on flow (i.e., connectivity) requirements, called source location problems, were introduced by Tamura et al [16], [17], and have recently received much attention from many authors (e.g., [1], [2], [3], [5], [9], [10], [12], [14], [15]). …”