32–approximation algorithm for two variants of a 2-depot Hamiltonian path problem

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Abstract

We consider two variants of a 2-depot Hamiltonian path problem and show that they have an algorithm with an approximation ratio of 32 if the costs are symmetric and satisfy the triangle inequality. This improves the 2-approximation algorithm already available for the problem.

Introduction

This article addresses the following 2-depot Hamiltonian path problem (2DHPP): Given a set of 2 distinct depots where a salesman is present at each depot, and a set of n2 destinations to visit, the objective of the problem is to

  • assign paths to the salesmen such that each destination is visited exactly once by any one salesman, and,

  • the sum of the cost of the paths of both the salesmen is minimized. The cost of a path is the total cost of the edges present in the path. In this problem, it is possible that only one of the salesmen is used for visiting all the destinations.

We also consider another variant of the above problem, denoted by 2DHPPv, where it is also required that each salesman must visit at least one destination in addition to all the constraints present in the 2DHPP. We are motivated to address the 2DHPP and 2DHPPv as it belongs to a class of multiple depot, routing problems that arises in applications [3], [4], [16] involving unmanned aerial vehicles (UAVs). The problems considered in this article are variants of these routing problems where there are two depots and the objective is to minimize the sum of the distances traveled by the two UAVs.

2DHPP and 2DHPPv are generalizations of a single depot, single salesman, Hamiltonian path problem (HPP) considered by Hoogeveen [9] and are NP-hard. Therefore, we are interested in developing approximation algorithms for the 2DHPP and 2DHPPv. An α-approximation algorithm [17] for problem P is an algorithm that

  • has a polynomial-time running time, and

  • returns a solution whose cost is within α times the optimal cost of problem P.

There are 2-approximation algorithms for several variants of the multiple depot, traveling salesman problems (TSPs) and HPPs in the literature [10], [12], [13], [14]. However, currently, there are no algorithms available for any multiple depot TSP or HPP that has an approximation ratio better than 2. In this article, we present 32-approximation algorithms for the 2DHPP and 2DHPPv when the costs are symmetric and satisfy the triangle inequality.

The TSP and HPP have received significant attention in the area of combinatorial optimization [8], [15]. The single, symmetric TSP has a well known 32-approximation algorithm by Christofides [5] when the costs satisfy the triangle inequality. This algorithm consists of the following steps:

  • 1.

    Find a minimum spanning tree.

  • 2.

    Find a minimum cost, perfect matching on all the odd-degree vertices of the minimum spanning tree.

  • 3.

    Add all the edges of the spanning tree and the perfect matching found in steps 1 and 2 to obtain an Eulerian graph, where the degree of each of the vertices is even. Using this graph, one can find an Eulerian tour that visits each of the edges exactly once.

  • 4.

    Shortcut the Eulerian tour to obtain a TSP tour that visits each of the vertices exactly once.

Hoogeveen generalized the above approximation algorithm for three variants of the HPP in [9]. The variants considered by Hoogeveen differed in the choice of the starting depot and the final destination of the HPP. In particular, Hoogeveen showed that there is a 32-approximation algorithm for a single HPP when the starting depot of the salesman is specified and the salesman is allowed to end his path at any of the destinations. The work in this article is aimed at developing similar approximation algorithms for variants of the 2-depot HPP.

In general, there are two subproblems when dealing with any multiple depot salesman problem. The first subproblem is the partitioning problem, which essentially requires finding a subset of destinations for each salesman to visit. Given the subset of vertices for a salesman to visit, the objective of the second subproblem, namely the sequencing problem, is to find an optimal sequence that produces the minimum cost path or tour. With respect to these two problems, consider the following algorithm for the 2DHPP:

  • 1.

    Solving the partitioning problem: Find a minimum cost forest with 2 trees spanning the depots and the destinations such that there is no path joining the two depot vertices. As there is no path joining the two depot vertices, there must be exactly one depot vertex in each tree of the minimum cost forest. The depot vertices present in the non-trivial trees (i.e., trees with at least one edge) correspond to the salesmen chosen to visit the destinations. There are algorithms available in the literature [12] to find such a minimum cost forest in polynomial time.

  • 2.

    Solving the sequencing problem: Double the edges in each non-trivial tree to get an Eulerian graph for each chosen salesman. Shortcut the edges in each Eulerian graph to obtain a path for each chosen salesmen.

It has been shown that the above algorithm has an approximation ratio of 2 when the costs are symmetric and satisfy the triangle inequality [12]. Similar 2-approximation algorithms for variants of the multiple depot TSPs and HPPs can also be found in [10], [13], [14]. Though there are several 2-approximation algorithms, it seems difficult to develop algorithms with an approximation ratio less than 2 for multiple depot TSPs and HPPs. The difficulty in dealing with multiple depot salesmen problems arises from the fact that both the partitioning problem and the sequencing problem are coupled. Currently, we do not know of algorithms in the literature that have an approximation ratio better than 2 for multiple depots TSPs or HPPs. In this article, we present a 32-approximation algorithm for variants of the HPP with two depots, i.e., the 2DHPP and 2DHPPv.

We first present an approximation algorithm for the 2DHPPv in Section 3. In Section 4, we then show that the proposed algorithm for the 2DHPPv has an approximation factor of 32. Using this approximation algorithm for the 2DHPPv and the Hoogeveen’s algorithm for the single depot HPP, we then propose a 32-approximation algorithm for the 2DHPP in Section 5.

Section snippets

Problem formulation

Let D={d1,d2} be the set of vertices representing the two distinct depots. There is one salesman located at each depot. Let U={1,2,3,,n} be the set of vertices denoting n destinations. We assume there are at least two destination vertices (n2). Let VDU. The edge (i,j) joining vertices i and j has a cost CijQ+ associated with it, where Q+ is the set of all positive rational numbers. Assume that all costs are symmetric, (i.e., Cij=Cji for all i,jV) and satisfy the triangle inequality (i.e.,

Approximation algorithm for the 2DHPPv

Consider the following algorithm, approxv, for generating a feasible solution, solf, for the 2DHPPv:

  • 1.

    Find a minimum cost constrained forest, F, such that (1) there are two trees in the forest (2) there is no path connecting the two depot vertices (i.e., there is exactly one depot in each tree), and (3) the degree of each of the depots is exactly equal to one. In Lemma 4.1, we later show that F can be found in polynomial time. Let the edges in the constrained forest be denoted by E(F). Note

Proof of the 32-approximation ratio of approxv

First, we show the number of steps required for approxv.

Lemma 4.1

The number of steps required for algorithm approxv is of O(n3) .

Proof

The computational complexity of approxv is dominated by its first two steps that include finding a minimum cost, constrained forest and finding a minimum cost, partial matching of given cardinality. Finding the minimum cost, constrained forest can be posed as a two matroid intersection problem. To see this, let V be the set of all the depots and the destinations (|V|=n+2),

Approximation algorithm for 2DHPP

In the 2DHPP, it is possible that only one of the salesmen is used for visiting all the destinations. Therefore, there are only three different scenarios in which two salesmen could be chosen: (1) only the first salesman is chosen for visiting all the destinations, (2) only the second salesman is chosen for visiting all the destinations, and (3) both the salesmen are chosen for visiting all the destinations. The algorithm for the 2DHPP uses the approximation algorithms already available to find

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