Approximation Algorithms for Some Postman Problems

Abstract

Approxtmatton algorithms for several NP-complete edge-covermg routing problems are presented and analyzed m terms of the worst-case ratio of the cost of the obtained solutmn to the cost of the optimum solutton A worst-case bound of 2 is proved for the mixed postman algortthm of Edmonds and Johnson, and a related algorithm for the mixed postman problem is shown also to have a worst-case bound of 2 A mixed strategy approach ts used to obtain a bound of ~ for the mixed postman problem A second mixed strategy algorithm, for the mtxed postman on a planar graph, ~s shown to have a worst-case bound of KEY WORDS AND PHRASES Chinese postman problem, mixed postman problem, rural postman problem, NPcomplete problems, polynomml-tlme approximation algorithm, heunsuc, worst-case performance bound, m,xed strategy CR CATEGORIES 5 25, 5 39, 8 3

Efficient Algorithms for Eulerian Extension

“…In contrast to Rural Postman, whose unweighted variant is NP-hard [12], we can show that EE and MEE are polynomial-time solvable. Altogether, our work complements and extends known results for WMEE with restricted weight function [11] and Rural Postman, for which mainly approximation, heuristic, and some polynomial-time algorithms for special cases are known [7,10].…”

“…For instance, it would be interesting to determine the parameterized complexity with respect to the parameter "number of weakly connected components" in a Weighted Multigraph Eulerian Extension instance. In this context, Orloff [17] observed that "the determining factor in the complexity of the problem seems to be the number (c) of connected components in the required edge set"; Frederickson [10] noted "the existence of an exact recursive algorithm that is exponential only in the number of disconnected components." However, it is doubtful that this meant fixed-parameter tractability with respect to c. In further future work, we also want to study the undirected and non-multigraph versions of WMEE.…”

“…With this equivalence, the NP-hardness of WMEE directly follows from the known NP-hardness result for Rural Postman [12]. Moreover, the fact that Rural Postman is solvable in polynomial time if the the set of required arcs is connected [10] directly implies that WMEE is solvable in polynomial time if the input is (weakly) connected.…”

A New View on Rural Postman Based on Eulerian Extension and Matching

“…A key issue in both problems is to determine the influence of the number c of connected components on each problem's computational complexity [10,17,18,23,29]. More precisely, c refers to the number of weakly connected components in the input graph for EE and the number of weakly connected components in the graph induced by the required arcs for RP.…”

“…If c = 1, then RP is efficiently solvable in polynomial-time [10]. Indeed, Frederickson [17,18] observed that, generally, RP is polynomial-time solvable when c is constant. However, c influences the degree of the polynomial in the running time of Frederickson's algorithm.…”