Two-terminal Series-parallel Graphs Research Articles

Aggregation approach for the minimum binary cost tension problem

The aggregation technique, dedicated to two-terminal series–parallel graphs (TTSP-graphs) and introduced lately to solve the minimum piecewise linear cost tension problem, is adapted here to solve the minimum binary cost tension problem (BCT problem). Even on TTSP-graphs, the BCT problem has been proved to be NP-complete. As far as we know, the aggregation is the only algorithm, with mixed integer programming (MIP), proposed to solve exactly the BCT problem on TTSP-graphs. A comparison of the efficiency of both methods and a heuristic is presented.

Real-time minimum vertex cover for two-terminal series-parallel graphs

In this paper we show how the tree contraction method can be applied to compute the cardinality of the minimum vertex cover of a two-terminal series-parallel graph. We then construct a real-time paradigm for this problem and show that in the new computational environment, a parallel algorithm is superior to the best possible sequential algorithm, in terms of the accuracy of the solution computed. Specifically, there are cases in which the solution produced by a parallel algorithm that uses p processors is better than the output of any sequential algorithm for the same problem, by a factor superlinear in p.

Minimum convex piecewise linear cost tension problem on quasi-k series-parallel graphs

Some hypermedia synchronization issues request the resolution of the minimum convex piecewise linear cost tension problem (CPLCT problem) on directed graphs that are close to two-terminal series-parallel graphs (TTSP-graphs), the so-called quasi-k series-parallel graphs (k-QSP graphs). An aggregation algorithm has already been introduced for the CPLCT problem on TTSP-graphs. We propose here a reconstruction method, based on the aggregation and the well-known out-of-kilter techniques, to solve the problem on k-QSP graphs. One of the main steps being to decompose a graph into TTSP-subgraphs, methods based on the recognition of TTSP-graphs are thoroughly discussed.

Minimum Dominating Trail Set for Two-Terminal Series Parallel Graphs

Abstract Given a graph G , the Minimum Dominating Trail Set (MDTS) problem consists in finding a minimum cardinality collection of pairwise edge-disjoint trails such that each edge of G has at least one endvertex on some trail. The MDTS problem is NP –hard for general graphs. In this paper an algorithmic approach for the MDTS problem on (two-terminal) series parallel graphs is presented.

Minimum‐cost dynamic flows: The series‐parallel case

AbstractA dynamic network consists of a directed graph with capacities, costs, and integral transit times on the arcs. In the minimum‐cost dynamic flow problem (MCDFP), the goal is to compute, for a given dynamic network with source s, sink t, and two integers v and T, a feasible dynamic flow from s to t of value v, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum‐cost maximum dynamic flow problem, where v is fixed to the maximum amount of flow that can be sent from s to t within time T and the minimum‐cost quickest flow problem, where is T is fixed to the minimum time needed for sending v units of flow from s to t. We first prove that both subproblems are NP‐hard even on two‐terminal series‐parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class 𝒢 of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). 𝒢 is a subclass of the class of two‐terminal series‐parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in 𝒢 in polynomial time. © 2004 Wiley Periodicals, Inc.

The searchlight guarding problem on weighted split graphs and weighted cographs

This paper addresses the searchlight guarding problem, which is an extension of so-called graph searching/guarding problem on a weighted, undirected graph G by considering the time-slot parameter in addition to the traditional building cost. Given a weighted, undirected graph G, suppose that there is a fugitive moving along the edges of G at an unknown speed. The task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive. Assume that it costs some building cost to arrange a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total costs of the vertices for S is minimized. Furthermore, for all sets of searchlights with a minimum building cost, the one with the minimum search time will be selected, that is, the time slots needed to spot the fugitive is furthermore required to be minimized. The searchlight guarding problem has been known to be NP-hard on weighted bipartite graphs. But it is linear-time-solvable on weighted trees, weighted interval graphs, and weighted two-terminal series-parallel graphs. In this paper, the searchlight guarding problem is first proved to be NP-hard on weighted split graphs. Next, a linear time optimal algorithm to solve the problem on weighted cographs is designed. The algorithm is divided into two phases: In the first phase, a set of searchlights with a minimum guarding cost is identified and the search directions of all edges are assigned by the dynamic programming strategy. To achieve the task involved in phase one, a new graph problem, the edge-direction assignment problem on weighted complete-bipartite graphs, is defined and solved using the greedy strategy. In the second phase, the search time slots of each edge are determined. Both phases take linear time. © 2000 John Wiley & Sons, Inc.

An efficient scheme to solve two problems for two-terminal series parallel graphs

An efficient scheme to solve two problems for two-terminal series parallel graphs

An optimal algorithm for solving the searchlight guarding problem on weighted two-terminal series-parallel graphs

In this paper, we consider a graph problem on a connected weighted undirected graph, called the searchlight guarding problem. Our problem is an extension of so-called graph searching/guarding problem by considering the time slot parameter in addition to the traditional building cost. Suppose that there is a fugitive who moves along the edges of the graph at any speed. We want to place a set of searchlights at the vertices to search the edges of the graph and capture the fugitive. It costs some building cost to place a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total costs of the vertices in S is minimized. If there is more than one set of searchlights with the minimum building cost, then find the one with the minimum searching time, that is, the time slots needed to capture the fugitive is the minimum. The problem is known to be NP-hard on weighted bipartite graphs, split graphs, and chordal graphs; and it is linear time solvable on weighted trees and interval graphs. In this paper, an algorithm is designed to solve the problem on weighted two-terminal series-parallel graphs. It works on the parsing tree structure of the given two-terminal series-parallel graph. The algorithm is divided into two phases. In the phase one, we first extract some useful properties of optimal solutions. Employing these properties, an algorithm is designed to find the set of searchlights with the minimum guarding cost and to assign the searching directions of all edges by the dynamic programming strategy. In the phase two, the searched time slots of all edges are determined by the breadth-first-search from the root of the parsing tree. The time complexities of both phases are linear. Thus, our algorithm is time optimal.

A note on the tour problems in two-terminal series-parallel graphs

In this paper, the tour problems in two-terminal series-parallel (TTSP) graphs is studied. A tour in a TTSP graph is a closed walk that passes a set of vertices. Those that pass every vertex at least once are called complete tour. The length of a tour is the summation of lengths of the edges in the tour. The shortest complete tour of a TTSP graph is a complete tour with minimum length. An O(¦E¦) time algorithm to find a shortest complete tour of a TTSP graph is presented, where ¦ E¦ is the number of edges in the graph. It is also shown that the bottleneck two-tour problem in a TTSP graph is NP-complete. The bottleneck two-tour problem in a TTSP graph is to find two tours such that each vertex of this TTSP graph must appear in exactly one of these two tours and the length of the longer tour of these two tours is minimized.

The most vital edges with respect to the number of spanning trees in two-terminal series-parallel graphs

A setE′ ofk edges in a multigraphG=(V,E) is said to be ak most vital edge set (k-MVE set) if these edges being removed fromG, the resultant graphG′=(V,E −E′) has minimum number of spanning trees. The problem of finding ak-MVE set for two-terminal series-parallel graphs is considered in this paper. We present anO (|E|) time algorithm for the casek=1, and anO(|V| k +|E|) time algorithm for arbitraryk.

Binary tree algebraic computation and parallel algorithms for simple graphs

In this paper we define the binary tree algebraic computation (BTAC) problem and develop an efficient parallel algorithm for solving this problem. A variety of graph problems (minimum covering set, minimum r-dominating set, maximum matching set, etc.) for trees and two terminal series parallel (TTSP) graphs can be converted to instances of the BTAC problem. Thus efficient parallel algorithms for these problems are obtained systematically by using the BTAC algorithm. The parallel computation model is an exclusive read exclusive write PRAM. The algorithms for tree problems run in O(log n) time with O( n) processors. The algorithms for TTSP graph problems run in O(log m) time with O( m) processors where n ( m) is the number of vertices (edges) in the input graph. These algorithms are within an O(log n) factor of optimal.

Parallel recognition and decomposition of two terminal series parallel graphs

In this paper, we develop a parallel recognition and decomposition algorithm for two-terminal series parallel (TTSP) graphs. Given a directed acyclic graph G in edge list form, the algorithm determines whether G is a TTSP graph. If G is a TTSP graph, the algorithm constructs a decomposition tree for G . Some interesting properties of the TTSp graphs are derived in order to facilitate fast parallel processing. The algorithm runs in O (log 2 n + log m ) time with O ( n + m ) processors on an exclusive read exclusive write PRAM where n ( m ) is the number of vertices (edges) in G . This algorithm is within a polylogrithmic factor of optimal.