Approximation Algorithms For The Min-Max Mixed Rural Postmen Cover Problem And Its Variants

With the development of information technology,the Arc Routing Problems have gradually penetrated into many emerging applications including medical care and autonomous driving,and have attracted the attention of many scholars.In this thesis,we study the multi-vehicle(or multi-postman)extension of the classical Mixed Rural Postman Problem(MRPP).The input of the problem consists of a positive integer k(representing the maximum number of postmen available)and a mixed graph G=(V,E,A),where V is the vertex set,E and A are respectively the(undirected)edge set and(directed)arc set.There are given edge subsets F(?)E and arc subsets H (?)A,and each edge or arc of the input graph G is associated with a non-negative weight(or length).The goal of the Min-Max Mixed Rural Postman Cover Problem(MRPCP)is to find at most k closed walks that traverse all edges and arcs in F and H such that the length of the longest walk is as short as possible.When k=1,it is the MRPP;when F=(?)and H=A,it is the Min-Max Stacker Crane Cover Problem(SCCP);when F=E and H=A,it is the Min-Max Mixed Chinese Postman Cover Problem(MCPCP).The goal of the Min-Max Mixed Rural Postman Walk Cover Problem(MRPWCP)is to find at most k walks that traverse all edges and arcs in F and H such that the length of the longest walk is as short as possible.If the input graph G satisfies the weakly symmetric condition,i.e.for every arc there is a parallel edge of no greater weight,this thesis proposes an algorithm for the MRPCP whose approximation ratio lies between 33/5 and 27/4 The specific approximation ratio depends on the ratio of the weight of H to that of F.When F=(?) and H=A,it is a 33/5approximation algorithm for the SCCP;when k=1,it is a 15/8-approximation algorithm for the MRPP.In addition,when the input graph G satisfies the weakly symmetric condition,we devise the first constant-factor approximation algorithms for the MRPWCP and the MCPCP,with ratios 5 and 4,respectively.Finally,the above results are extended to the case where the input graph satisfies conditions that are more relaxed than the weakly symmetric condition.