It is well-knawn that the tight lower bounds determine the effectiveness of the branch and bound method fbr the NP-hard problems, In this paper, we present a new lower bounding procedure fbr the capacitated arc routing problem (CARP), one of the arc routing preblems, They give the tight lower bounds aRd it is easy to develop an exact algorithm using their network structures. 1 Introduction The routing problems have been studied by many researchers for more decades. The arc routing problem (ARP) is one of the routing problems which focuses on arcs in a network. This problem includes the well-known "Chinese postman problem (CPP)", CPP is a problem of covering all arcs in a network while minimizing the total distance cost traveled, CPP was presented by Meiko-Kwan [9] and solved polynomially by Edmonds afid Johnson [5] based on the minimum-cost perfect matching problem (MCPM) in the general graph. CPP is said to be attractive, because it is an exceptionally well-solved problem in ARP alld has a number of applications like mai1 deliveiy [5]. On the other hand, since CPP is a simple structured pioblem, there are many problems in ARP to which CPP algoTithmis directly inapplicable, For example, the routing of street sweepers, snow plows [41, household refuse collection vehic}es, the spra(ying of roads with saltgrit to prevent ice formation, the inspection of electric power lines [15] gas or oil pipelines for faults and so on. In this paper, we consider one of these problems, so called the capacitated arc routing probgem (CARP). As is mentioned in [71, CARP includes such related problems as the traveling salesman problem (TSP), the Chinese postman prob}em (CPP) [2,5], the rural postman problem (RPP) [3], the capacitated Chinese postman problem (CCPP), the vehicle routiRg problem (VRP) [6,10] and the general routing problem (GRP) [11,i2]. Golden and Wong [7] showed that CARP is a NP-hard problem. Thus recent reseamchers have focused their effbrt on deve}oping and testing heuristics. Also the lower bounding procedures [1,7,13] have been developed to estimate the eficiency of the heuristics. These lower bounds can be ebtained eficiently by solving MCPM on simple structured networks. However, since they relax the capacity constraint, i.e., one of the constraints in CARP, we point out that bounds are not tight. Moreover, it is hard to construct an exact solution through a bTallch and bound method using their network structures.
