A New Approach for Solving the Generalized Traveling Salesman Problem

The generalized traveling problem (GTSP) is an extension of the classical traveling salesman problem. The GTSP is known to be an NP-hard problem and has many interesting applications. In this paper we present a local-global approach for the
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  A New Approach for Solving the GeneralizedTraveling Salesman Problem P.C. Pop 1 , O. Matei 2 , C. Sabo 1 1 Dept. of Mathematics and Computer Science, North University of Baia Mare,Str. Victoriei, 430122, Baia Mare, Romania, cosmin 2 Dept. of Electrical Engineering, North University of Baia Mare,Str. V. Babes, 430083, Baia Mare, Romania Abstract.  The generalized traveling problem (GTSP) is an extensionof the classical traveling salesman problem. The GTSP is known to be anNP-hard problem and has many interesting applications. In this paperwe present a local-global approach for the generalized traveling salesmanproblem. Based on this new approach we describe a novel integer pro-gramming formulation of the GTSP and as well a hybrid metaheuristicalgorithm for solving the problem using genetic algorithms. Computa-tional results are reported for Euclidean TSPlib instances and comparedwith the existing ones. The obtained results point out that our hybridalgorithm is an appropriate method to explore the search space of thiscomplex problem and leads to good solutions in a reasonable amount of time. Keywords : generalized traveling salesman problem, hybrid algorithms,genetic algorithms, integer programming. 1 Introduction The classical traveling salesman problem can be generalized in a natural way byconsidering a related problem relative to a given partition of the nodes of thegraph into node sets (clusters), while the feasibility constraints are expressed interms of the clusters, i.e. each of the clusters has to be visited once and onlyonce (i.e. exactly once). In the literature, there are considered two versions of the problem: –  one in which we are interested in finding the shortest closed tour visitingexactly one node from each cluster. This problem is called the  generalized traveling salesman problem   (GTSP) and have been introduced independentlyby Henry-Labordere [6], Srivastava et al. [22] and Saskena [19]; –  the second one is the problem of finding the shortest closed tour including  at least   one vertex from each cluster. This version of the problem was introducedby Laporte and Nobert [8] and by Noon and Bean [12].  2In the present paper we confine ourselves to the problem of choosing exactlyone node from each of the clusters. The TSP is a special case of the GTSP whereeach cluster consists of exactly one node.The GTSP is NP-hard, as it reduces when each cluster consists of exactlyone node to the traveling salesman problem which is known to be an NP-hardproblem. The GTSP has several applications to location problems, planning,postal routing, logistics, manufacture of microchips, telecommunication prob-lems, railway optimization, etc. More information on the problem and its appli-cations can be found in Fischetti, Salazar and Toth [3,4], Laporte, Asef-Vaziri and Sriskandarajah [9], etc.A lot of attention was payed by the researchers for solving the GTSP: therehave been proposed several transformations of the GTSP into TSP, an efficientexact algorithm have been proposed by Fischetti, Salazar and Toth [4] based on abranch-and-bound algorithm that solved the problem to optimality for instanceswith up to 89 clusters and 442 nodes, etc.The difficulty of obtaining optimal solutions for the GTSP has led to thedevelopment of several heuristic and metaheuristic algorithms: an efficient com-posite heuristic [18], reinforcing ant colony system [13], a random key genetic algorithm [21], variable neighborhood search [7], memetic algorithms [5,2], ant colony algorithms [23], etc.Recently, several hybrid algorithms have been introduced to solve efficientlycombinatorial optimization problems based mainly on combination of meta-heuristic algorithms or combination of exact methods with metaheuristic al-gorithms.Hybrid algorithms exploit the good properties of different methods by ap-plying them to optimization problems they can efficiently solve. For example,searching the solution space of an optimization problem is efficient when theproblem has many solutions. In the case of combinatorial optimization problemsthe hybrid algorithms are design to achieve a vast exploration of the search space,by escaping from local optima and intensifying at promising solutions regions.The aim of this paper is to describe a new approach to the GTSP basedon distinguishing between  global connections   (connections between clusters) and local connections   (connections between nodes from different clusters). This ap-proach leads to a new integer linear programming formulation to the GTSP andin combination with a genetic algorithm to an efficient hybrid algorithm for sol-ving the problem, which is competitive with the other modern heuristics in termsof computing time and quality solution. Computational results for benchmarksproblems are reported. 2 Definition of the GTSP Let  G  = ( V,E  ) be an  n -node undirected graph whose edges are associated withnon-negative costs. We will assume without loss of generality that the graph  G is a complete graph (if there is no edge between two nodes, we can add it withan infinite cost).  3Let  V   1 ,...,V   m  be a partition of   V    into  p  subsets called  clusters   (i.e.  V    = V   1  ∪  V   2  ∪  ...  ∪  V   m  and  V   l  ∩  V   k  =  ∅  for all  l,k  ∈ { 1 ,...,m } ). We denote the costof an edge  e  =  { i,j } ∈  E   by  c ij . The  generalized traveling salesman problem  asks for finding a minimum-cost tour  H   spanning a subset of nodes such that H   contains exactly one node from each cluster  V   i ,  i  ∈ { 1 ,...,m } . Therefore theGTSP involves the following two related decisions: –  choosing a node subset  S   ⊆  V    , such that  | S   ∩  V   k |  = 1, for all  k  = 1 ,...,m . –  finding a minimum cost Hamiltonian cycle in the subgraph of   G  induced by S  .We will call such a cycle a  generalized Hamiltonian tour  . An example of a generalized Hamiltonian tour for a graph with the nodes partitioned into 6clusters is presented in the next figure. Fig.1.  Example showing a generalized Hamiltonian in the graph  G  = ( V,E  ) The GTSP is NP-hard, as it reduces when each cluster consists of exactlyone node ( | V   i |  = 1,  ∀  i  = 1 ,...,m ) to the traveling salesman problem which isknown to be an NP-hard problem. 3 The local-global approach to the Generalized TravelingSalesman Problem Let  G ′ be the graph obtained from  G  after replacing all nodes of a cluster  V   i with a supernode representing  V   i . We will call the graph  G ′ the global graph.For convenience, we identify  V   i  with the supernode representing it. Edges of thegraph  G ′ are defined between each pair of the graph vertices  V   1 ,...,V   m .The local-global approach was introduced by Pop [14] in the case of thegeneralized minimum spanning tree problem. Since then this approach was suc-cessfully applied for in order to obtain exact algorithms, strong mixed-integerprogramming formulations, heuristic and metaheuristic algorithms for severalgeneralized network design problems, see [15,16,7]. The local-global approach to the GTSP aims at distinguishing between  global connections   (connections between clusters) and  local connections   (connections  4between nodes from different clusters). As we will see, given a sequence in whichthe clusters are visited (global Hamiltonian tour) it is rather easy to find thecorresponding best (w.r.t. cost minimization) generalized Hamiltonian tour.There are several generalized Hamiltonian corresponding to a global Hamil-tonian tour. Between these generalized Hamiltonian tours there exists one calledthe best generalized Hamiltonian tour (w.r.t. cost minimization) that can bedetermined using one of the methods that we are going to describe next.Given a sequence ( V   k 1 ,...,V   k m ) in which the clusters are visited, we want tofind the best feasible Hamiltonian tour  H  ∗ (w.r.t cost minimization), visitingthe clusters according to the given sequence. This can be done in polynomialtime, by solving  | V   k 1 |  shortest path problems as we will describe below.We construct a layered network, denoted by LN, having  m  + 1 layers corre-sponding to the clusters  V   k 1 ,...,V   k m  and in addition we duplicate the cluster  V   k 1 .The layered network contains all the nodes of   G  plus some extra nodes  v ′ for each v  ∈  V   k 1 . There is an arc ( i,j ) for each  i  ∈  V   k l  and  j  ∈  V   k l +1  ( l  = 1 ,...,m − 1),having the cost  c ij  and an arc ( i,h ),  i,h  ∈  V   k l , ( l  = 2 ,...,m ) having cost  c ih .Moreover, there is an arc ( i,j ′ ) for each  i  ∈  V   k m  and  j ′ ∈  V   k 1  having cost  c ij ′ . V k V k 1  V k  V k  V k 2 3 p 1 v  v’  ... Fig.2.  Example showing a Hamiltonian tour in the constructed layered network LN For any given  v  ∈  V   k 1 , we consider paths from  v  to  v ′ ,  v ′ ∈  V   k 1 , that visitsexactly one node from each cluster  V   k 2 ,...,V   k m , hence it gives a feasible Hamil-tonian tour.Conversely, every Hamiltonian tour visiting the clusters according to thesequence ( V   k 1 ,...,V   k m ) corresponds to a path in the layered network from acertain node  v  ∈  V   k 1  to  v ′ ∈  V   k 1 .Therefore, it follows that the best (w.r.t cost minimization) Hamiltonian tour H  ∗ visiting the clusters in a given sequence can be found by determining all theshortest paths from each  v  ∈  V   k 1  to the corresponding  v ′ ∈  V   k 1  with the propertythat visits exactly one node from each of the clusters ( V   k 2 ,...,V   k m ).The overall time complexity is then  | V   k 1 | O ( | E  | +log n ), i.e.  O ( n | E  | + nlogn )in the worst case, where by  | E  |  we denoted the number of edges. We can reducethe time by choosing  | V   k 1 |  as the cluster with minimum cardinality.Notice that the above procedure leads to an  O (( m − 1)!( n | E  | + nlogn )) timeexact algorithm for the GTSP, obtained by trying all the ( m − 1)! possible clustersequences. So, we have established the following result:  5 Theorem 1.  The above procedure provides an exact solution to the generalized traveling salesman problem in   O (( m  −  1)!( n | E  |  +  nlogn ))  time, where   n  is the number of nodes,  | E  |  is the number of edges and   m  is the number of clusters in the input graph. Clearly, the algorithm presented, is an exponential time algorithm unless thenumber of clusters  m  is fixed.Given a global Hamiltonian tour, the corresponding best generalized Hamil-tonian tour can be determined by solving a linear integer program as we willdescribe in what it follows.We introduce the following binary variables  x e  ∈ { 0 , 1 } ,  e  ∈  E   and  z i  ∈ { 0 , 1 } , i  ∈  V    to indicate whether an edge  e  respectively a node  i  is contained in theHamiltonian tour, and the variables  y ij , ( i,j  ∈ { 1 ,...,m } ) to describe the globalconnections. So  y ij  = 1 if cluster  V   i  is connected to cluster  V   j  and  y ij  = 0otherwise. We assume that  y  represents a Hamiltonian tour. The convex hull of all these  y -vectors is generally known as the Hamiltonian tour polytope on theglobal graph  G ′ .Following Miller  et al.  [11] this polytope, denoted by  P  TSP  , can be repre-sented by the following polynomial number of constraints: m  j =1 ,j  = i y ij  = 1 ,  ∀ i  ∈ { 1 ,...,m }  (1) m  i =1 ,i  = j y ij  = 1 ,  ∀  j  ∈ { 1 ,...,m }  (2) u i  −  u j  + ( m  −  1) y ij  ≤  m −  2 , ∀ i,j  ∈ { 2 ,...,m } , i   =  j  (3)1  ≤  u i  ≤  m  −  1 ,  ∀ i  ∈ { 2 ,...,m }  (4)where the extra variables  u i  represent the sequence in which city  i  is visited, i   = 1.The first two constraints guarantee that each of the nodes (cities) is visitedexactly once. The constraints denoted by (3) ensure that the solution containsno subtour on a set of nodes  S   with  | S  | ≤  m − 1 and hence, no subtour involvingless than  m  nodes and finally the constraints denoted by (4) ensure that the  u i variables are uniquely defined for any feasible tour.If the vector  y  describes a Hamiltonian tour on the global graph  G ′ , then thecorresponding best (w.r.t. minimization of the costs) generalized Hamiltoniantour can be obtained by solving the following 0-1 programming problem:min  e ∈ E   c e x e s.t. z ( V   k ) = 1 ,  ∀  k  ∈  K   =  { 1 ,...,m } x ( V   l ,V   r ) =  y lr ,  ∀  l,r  ∈  K   =  { 1 ,...,m } ,l   =  rx ( i,V   r )  ≤  z i ,  ∀  r  ∈  K, ∀ i  ∈  V    \  V   r x e ,z i  ∈ { 0 , 1 } ,  ∀  e  = ( i,j )  ∈  E, ∀ i  ∈  V,
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