A bilevel programmingapproach to the travellingsalesman problem

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1 Available online at Operations Research etters Operations Research etters A bilevel programmingapproach to the travellingsalesman problem Patrice Marcotte a;, Gilles Savard b,frederic Semet c a DIRO, Universite de Montreal and CRT, CP 6128, Succursale Centre-Ville, Montreal, QC, Canada H3C 3J7 b Ecole Polytechnique de Montreal and GERAD, C.P. 6079, Succursale Centre-Ville, Montreal, QC, Canada H3C 3A7, Canada c AMIH, Universite de Valenciennes, Valenciennes Cedex 9, France Received 17 December 2002; accepted 8 August 2003 Abstract We show that the travellingsalesman problem is polynomially reducible to a bilevel toll optimization program. Based on natural bilevel programming techniques, we recover the lifted Miller Tucer Zemlin constraints. Next, we derive an On 2 multicommodity extension whose P relaxation is comparable to the exponential formulation of Dantzig, Fulerson and Johnson. c 2003 Elsevier B.V. All rights reserved. Keywords: Bilevel programming; Pricing; Travelling salesman problem 1. Introduction The travellingsalesman problem TSP is a well-researched problem whose interest lies beyond the Icosian game or the Knight s tour puzzle. Due to its attractive name, practical importance and theoretical intractability, the TSP has attracted a lot of attention from the Operations Research and Mathematical Programming communities see [8]. In this short wor, we uncover a new facet of this problem, namely its close relationship with a toll collection problem modelled as a bilinear bilevel program. Not only is this analogy interesting in itself, but it also yields tight lower bounds for the TSP. Research partially supported by NSERC and MITACS Canada, and FCAR Quebec, and region Nord Pas de Calais. Correspondingauthor. Tel.: ; fax: address: marcotte@iro.umontreal.ca P. Marcotte. The classical TSP consists in determininga least cost Hamiltonian circuit in a directed graph. Several mathematical programming formulations, either exponential or polynomial in terms of the number of nodes n, have been proposed for the TSP see [7]. In a seminal paper, Dantziget al. [2] proposed an exact formulation involvingan exponential number of constraints. ater, On 3 integer programming formulations yieldingthe same linear relaxation bounds were obtained by Claus [1] and Wong[11], among others. In this paper we propose On 2 formulations whose linear programming relaxations provide lower bounds of quality comparable to that obtained by Dantziget al. [2]. The basic formulation is derived from a bilevel programming model of revenue imization see [6] which allows, in a natural fashion, to derive the lifted Miller Tucer Zemlin MTZ constraints obtained by Desrochers and aporte [3]. Also, we strengthen the basic formulation by introducing /$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi: /j.orl

2 commodities and demonstrate, through numerical experiments, the inuence of the number of commodities on the quality of the lower bound. P. Marcotte et al. / Operations Research etters xa + ya xa + ya=b i a i a i + i N; K 2. A toll optimization problem A bilevel program describes a hierarchical structure where the feasible set of a leader is contingent on the follower s optimal reaction to the leader s decisions. In [6], abbe et al. formulated the toll optimization problem TOP as a bilevel program where the leader upper level sets tolls on a subset of arcs of a transportation networ, while the followers lower level travel on shortest paths with respect to a generalized cost. More precisely, let us consider a multi-commodity networ where each commodity K is associated with an origin destination pair o;d of a transportation networ G with node set N and arc set A, the latter beingpartitioned into the subset A 1 of toll arcs and the complementary subset A 2 of toll-free arcs. With each arc a of A 1 is associated a generalized travel cost composed of a xed part c a representingthe unit travel cost and a toll T a. Any toll-free arc a of A 2 bears a xed unit travel cost d a. We denote by {n } K the demand for commodity, a negative demand corresponding to supply, and introduce the nodal demands n if i = o; b i = n if i = d; 0 otherwise: At the lower level, the variable xa respectively ya represents the number of commodity users that travel on arc a A 1 respectively on arc a A 2. For a given toll vector T, we assume that users minimize their individual generalized travel costs, i.e., users are assigned to shortest paths lining their respective departure and arrival nodes. The correspondinglower level mathematical program, parameterized in the toll vector T, is min x;y K c a + T a xa + d a ya a A 1 a A 2 x a 0 K a A 1 ; y a 0 K a A 2 ; 1 where i + respectively i denotes the set of outgoingrespectively incoming arcs incident to node i. A set of revenue-imizingtolls is then obtained by solvingthe program T T a xa a A 1 K Ta min 6 T a 6 Ta a A 1 ; 2 where it is understood that the ow variables are optimal solutions of the lower level program 1. Whenever the lower level solution is not unique, we assume that ties are broen in the leader s favour. This is in accordance with usual bilevel practice, and serves our theoretical purpose. The membership of TOP in NP follows from a property shared with linear programming: if TOP has an optimal solution, then there exists an optimal solution which is a vertex of a suitably dened polyhedron P. In other words, TOP can be viewed as a combinatorial optimization problem dened over the vertices of P, rather than a continuous nonconvex mathematical program. This combinatorial problem has been shown to be NP-hard by abbe et al. [6]. 3. Reformulating the TSP as a TOP In this section, we formulate the TSP as a single-commodity TOP. This is achieved in two steps: rst, we adapt the reduction from the Hamiltonian path problem HPP to TOP proposed in [6] tothe Hamiltonian circuit problem HCP; next, we extend this reduction technique to the TSP, taingthe cost structure into account Equivalence of the HCP and TOP Given a directed graph G =N; A, the HCP consists in determiningan elementary circuit that passes

3 242 P. Marcotte et al. / Operations Research etters through each node of N. We dene a TOP model by specifying its underlyinggraph, its cost structure, lower and upper bounds on tolls. Since TOP involves paths rather than circuits, it is convenient to transform the HCP into an equivalent HPP. This is achieved by duplicatingnode 1 its image is denoted n+1 and replacing, for i distinct from either the origin or destination node, arc i; 1 A by arc i; n + 1. ooingfor a Hamiltonian circuit in the original graph is equivalent to identifying a Hamiltonian path from node 1 to node n +1. et us identify the arc set A 1 of the modied graph with the set of toll arcs, and let us incorporate a single toll-free arc 1;n+ 1. The resultinggraph is denoted by G =N; A with N = N {n +1} and A = A 1 {1;n+1}. We associate with the origin destination pair 1;n+1 a unit demand, with each toll arc i; j A 1 a xed cost of 1, and set the cost of the toll-free arc 1;n+1ton. Finally, a lower bound of 2 is imposed on all tolls, and the upper bound is set to +. Our aim is to show that there exists a Hamiltonian circuit in G if and only if the optimal revenue of the above TOP is equal to 2n. First, let us note that the revenue cannot exceed 2n. Indeed, all arc costs inclusive of toll beingpositive, the lower level solution is an elementary path. Since xed costs on toll arcs are set to 1 and that the cost of the shortest path is less or equal to n there exists a toll-free path from node 1 to n + 1 of cost n, an upper bound on the revenue associated with an optimal path using l arcs is given by n l 1 = n + l: This bound corresponds to the dierence between the cost of the shortest toll-free path and the xed cost of any path using l arcs. Hence, a revenue of 2n can only be generated by a shortest path using exactly n arcs, i.e., a Hamiltonian path. Conversely, assume that there exists a Hamiltonian path p between nodes 1 and n + 1. For each arc i; j on p, set the toll variables T ij to their common lower bound 2, and to n 1 on arcs outside p. Under this cost structure, the shortest path is clearly p and the correspondingrevenue is 2n. Note that the marginal benet raised on any given toll arc decreases with increasingtoll value. For instance, if the toll is set at its lower bound 2, the arc cost is equal to 1; this leads to a benet/cost ratio of 2, while a toll set at 3 results in a ratio of 1.5. Intuitively, it is to the leader s advantage to set the toll as low as possible, thus inducingthe users to travel on the longest possible path, e.g., a Hamiltonian path. Of course, all tolls on the arcs that are not part of the Hamiltonian path should be set at a value high enough to discourage their use. The argument relies heavily on the negativity of the xed costs and on the lower bound on tolls, which prevents the benet/cost ratio of growing innite A TOP formulation of the TSP By perturbingslightly the cost and bound structure in the TOP formulation of the HPP, one may preserve the benet/cost ratio principle, while favouringthe least cost paths. Accordingly, the users will travel on a longhamiltonian path of lowest cost, i.e., an optimal TravellingSalesman tour. In the perturbed problem, the xed cost of toll arc i; j is set to 1+c ij = and the lower bound on the correspondingtoll is set at 2 c ij =, where is an upper bound on the cost of an optimal tour, for instance = n i;j A {c ij }. This yields the bilevel program TOP TSP : min x T T ij 2 c ij i;j A 1 +nx 1n+1 j j;i A x ji i;j A 1 T ij x ij i; j A 1 ; 1+ c ij j i;j A + T ij x ij x ij 1 if i =1; = 1 if i = n +1; x ij 0 i; j A:

4 P. Marcotte et al. / Operations Research etters We now show that the optimal solution of TOP TSP yields tolls that induce an assignment of lower level ows to a shortest Hamiltonian path. To this aim, let pt denote an optimal path for a given toll vector T. If pt uses l arcs, we have that i;j pt 1+T ij + c ij which yields the inequalities n + l revenue = i;j pt i;j pt =2l 1 6 n; 3 T ij 2 c ij i;j pt c ij 2l 1: It follows that a revenue larger than 2n 1 can only be achieved by inducingthe follower to travel on a path of length n, i.e., a Hamiltonian path p H T. In order that the cost of this path be competitive with the cost of the toll-free path, inequality 3 must hold. For a given Hamiltonian path p H T, the imal revenue is obtained when the equality T ij =2n 1 c ij i;j p HT i;j p HT holds. This can easily be achieved by settingthe tolls on the arcs of p H at their lower bound, and the remainingtolls at. The optimal path corresponds, clearly, to a Hamiltonian path p of minimal cost i;j p c ij in the graph G. In the original graph G, p corresponds to a Hamiltonian tour of minimum cost, i.e., a solution of the TSP. One possible choice for the toll vector T is, as indicated above Tij = 2 c ij if i; j p ; if i; j p : 4 The choice of the constant ensures that any alternative path has a cost higher than p A mixed integer formulation for the TSP In this section, we reformulate TOP TSP as a mixed integer program MIP by performing standard operations on a bilevel program and obtain, as a consequence, the lifted version of the Miller Tucer Zemlin formulation of the TSP proposed by Desrochers and aporte [3]. First, we replace the lower-level shortest path program by its optimality conditions, to derive the equivalent single-level program: MIP : T; x j j;i A i;j A 1 T ij x ij x ji j i;j A x ij 1 if i =1; = 1 if i = n +1; j i 6 1+T ij + c ij i; j A 1 ; n n; 1+T ij + c ij + i j x ij =0 i; j A 1 ; n + 1 n+1 x 1n+1 =0; T ij 2 c ij i; j A 1 ; x ij 0 i; j A; 5 where, without loss of generality, we set 1 = 0, i.e., i represents the shortest distance from node 1 to i.at an optimal solution, we obviously have n+1 = n and x 1n+1 =0. Based on the construction of abbe et al. [6] and on the fact that the ow variables x ij are binary valued for an extremal optimal path, we are able to propose a stronger linearization of the complementarity constraints without any need for additional variables. First, note that the choice of T specied in 4 implies x ij = x ij T ij 1+ c ij at the optimal solution. This allows to rewrite the rst complementarity term of system 5 as 1 + i j x ij =0:

5 244 P. Marcotte et al. / Operations Research etters If x ij = 1, this is equivalent to the reverse inequalities j i 6 x ij ; 6 j i x ij : 7 Symmetrically, if x ji = 1, we obtain i j 6 x ji 8 and i j x ji : 9 Since x ij and x ji cannot both be nonzero, we can merge 7 and 8 into the single constraint i j 6 x ji x ij + M1 x ij x ji ; 10 for some suitably large constant M. et us consider the case where i n + 1 and j 1. Since i j cannot exceed n 2, we can set M = n 2 to derive the valid inequality i j 6 n 2+1 nx ij +3 nx ji : 11 Consideringconstraints 6 and 9 instead of 7 and 8 leads to the same constraint. et us now consider the case where i =1orj = n +1.Ifx 1j = 1 and since 1 = 0, there holds j 0 6 x 1j ; 12 j 0 x 1j : 13 Similarly, if x jn+1 = 1 and since n+1 = n, we obtain the inequalities n j 6 x jn+1 ; 14 n j x jn+1 : 15 Consideringconstraints 12 and 15, we obtain j 6 x 1j +n 1x jn+1 + M1 x 1j x jn+1 : 16 Noticingthat j 6 n 2ifx 1j = 0 and x jn+1 =0,we derive j 6 n 2+3 nx 1j + x jn+1 : 17 If we consider constraints 13 and 14, and following the same development, we obtain j n 3x jn+1 x 1j +2: 18 Finally, since the optimal toll vector induces a lower-level tour, one can replace the ow conservation constraints of MIP by the equivalent degree constraints 19 and 20. This yields the mixed-integer formulation MIP-TSP : x i;j A 1 2 c ij x ij min x i;j A 1 c ij x ij x ij =1 i N; 19 j i;j A 1 x ji =1 i N; 20 j j;i A i j 6 n 2+1 nx ij +3 nx ji i; j A 1 ; 21 j 6 n 2+3 nx 1j + x jn+1 j {2;:::;n}; 22 j n 3x jn+1 x 1j +2 j {2;:::;n}; 23 x ij {0; 1} i; j A 1 : This single-commodity reformulation of the TSP is nothingbut the lifted formulation of MTZ derived by Desrochers and aporte [3]. Constraint 21 corresponds to the lifted subtour elimination constraint of MTZ, and denes a facet of the MTZ polytope. Constraints 22 and 23, which are obtained by lifting the bound constraints on the potentials i, have been proved to be facet-deningfor MTZ by Driscoll [4] A multi-commodity extension In this section, we introduce a multi-commodity TOP reformulation of the TSP, where a tour is partitioned into contiguous simple paths, each path being associated with a commodity K. More precisely, let us denote by o the origin node of path commodity and by d the destination node of path. Assumingthat we now the orderingof the commodities on some optimal tour, we can write o+1=d for =1;:::; K 1 and d K =o1. To construct a TOP, we associate with each commodity a variable

6 P. Marcotte et al. / Operations Research etters yo;d associated with the toll-free arc o;d. In contrast with the single-commodity case, the xed cost of toll-free arcs are not nown a priori. et i denote the potential associated with commodity. In order that the formulation be valid, the followingproperties must hold: 1. for each commodity index, the potential i is equal to the number of arcs from the origin o to the destination d in the optimal tour; 2. the costs w of the toll-free arcs must sum up to n; 3. each toll arc must be used once and only once. Followingthe same line of reasoningas for the single-commodity case, we derive the formulation: K TOP-TSP : T; x;y;w K i;j A T ij 2 c ij w = n; K K j i;j A min x;y T ij x ij; i; j A; x ij 6 1 i N; w 0 K; K i;j A 1+ c ij + K w y o;d j j;i A x ji j i;j A x ij + T ij x ij yo;d 1 if i = o; = yo;d +1 if i = d; x ij 0 i; j A K; y o;d 0 K: The arguments developed in Section 3.2 to validate the TOP TSP model can be extended to the multi-commodity formulation. For commodity, let p T denote the shortest path associated with a toll vector T. Ifp T uses l arcs, we have that i;j p T 1+T ij + c ij 6 w ; which yields the inequalities w + l T ij i;j p T =2l 1 i;j p T i;j p T c ij : 2 c ij Summingover the commodity indices, we obtain n + l T ij 2 c ij i;j p T i;j p T =2l 1 K i;j A i;j p T c ij 2l 1; where l is the total number of toll arcs used by all commodities. These inequalities, together with arguments similar to those of Section 3.2, allow us to arm that an optimal tour p can be recovered by setting 2 c ij if i; j p ; Tij = otherwise: Writingdown the lower-level optimality conditions, eliminatingtoll-free variables yo;d which must vanish at the optimal solution and the associated w variables, linearizingthe complementarity constraints and determiningtight bounds on, we derive a mixed-integer formulation. Note that, in contrast with the single-commodity case, the value of o and d cannot be set a priori since the number of nodes on each commodity path is unnown a priori. However, we can either x the value of o to zero or the value of d to n K + 1. Fixingthe values of o to zero yields the single-level problem 2 c ij xij T; x;y K j i;j A x ij =1 i N;

7 246 P. Marcotte et al. / Operations Research etters j j;i A x ij j i;j A i j 6 n K 1 1 if i = o; xji = 1 if i = d; 24 i j 6 n K 1+ K n K x ij + K +2 n K i and j neither an origin or x ji destination node; 31 + K nx ij + K +2 nx ji i and j distinct from o or d; K; 25 j 6 n K 1 + x jd n K 2x oj j distinct from o or d; K; 26 j 2 2x jd x oj j N; jdistinct from o or d; K; 27 xij {0; 1} i; j A K: On the other hand, had we xed d to n K + 1, then constraints 26 and 27 would have been replaced by j 6 n K 1 + xjd +2xoj; 28 j n K 2xjd xoj +2: 29 One can reduce the number of variables by observing that each commodity uses distinct paths and never visit twice the same node. Hence the variables i can be set to arbitrary values in particular 0 at the nodes outside its path, or simply discarded. Finally, since an arc is used by at most one commodity, we can lift constraints to obtain the streamlined formulation: 2 c ij xij T; x;y K i;j A K j i;j A j j;i A x ij =1 i N; xji 1 if i=o; xij = 1 if i=d; j i;j A 30 j 6 n K 1 + K x jd n K 2 K x oj j 2 2 K j neither an origin or destination node; 32 x jd K x oj j neither an origin or destination node; 33 xij {0; 1} i; j A; K: Alternatively, we could have lifted constraints 28 and 29. Remar. The a priori nowledge of the ordering of commodities in K might seem a strong assumption. However, in the symmetric case, we can select any three nodes to construct a 3-TOP TSP formulation of the original problem. Indeed, the symmetric cost structure implies that all six permutations are equivalent. Moreover, in the Euclidean case, it has been proved by Flood [5] that the optimal tour visits vertices of the convex envelope in the natural cyclic order, e.g., clocwise. Hence, if the nodes in K belongto the set of such vertices, only one K -TOP TSP has to be solved to nd a solution to the original TSP. 4. Numerical experiments In order to assess the quality of our formulations, we solved the linear programming relaxations of TOP TSP and 3-TOP TSP. Our testbed is composed of three problem classes: random Euclidean, symmetric and asymmetric. The last two sets of problems are taen from the TSPIB library [9].

8 P. Marcotte et al. / Operations Research etters Table 1 50-node problems 1-TOP GAP % 3-TOP GAP % DFJ MTZ + vs. DFJ vs. DFJ Average Std. dev Table 2 Euclidean TSPIB problems Problem 1-TOP TSP GAP % 3-TOP TSP GAP% DFJ GAP % OPT MTZ + vs. OPT vs. OPT vs. OPT gr gr gr bayg bays dantzig gr eil berl brazil st eil pr gr roa rob roc rod roe eil gr bier gr roa rob roa rob Average Std. Dev

9 248 P. Marcotte et al. / Operations Research etters Table 3 Asymmetric TSPIB problems Problem 1-TOP TSP GAP% SD GAP% 3-TOP TSP GAP% vs. OPT MTZ + vs. OPT vs. OPT vs. OPT br p ry48p The random problems consist of Euclidean TSPs of size 50 complete graphs where nodes, rounded to the nearest integer, are generated according to the procedure proposed by Desrochers and aporte [3]: c ij =[x i x j 2 +y i y j 2 ] 1=2 ; x i ;y i U[0; 50] 2 for i j: In Table 1 we present the linear relaxations for the TOP TSP formulation equivalent to the lifted version MTZ + of Miller, Tucer and Zemlin s formulation, the 3-TOP TSP formulation, and the classical bound of Dantzig, Fulerson and Johnson s DFJ. In Table 1, the heading GAP refers to the dierence between the value of the DFJ relaxation and the value of the relaxation lower bound, expressed in percentages. In Tables 2 and 3, GAP is computed with respect to the optimal solution OPT. For 3-TOP TSP, the nodes o, =1; 2; 3 have been selected so as to imize the area of the triangle with vertices o1, o2 and o3. Next, we consider Euclidean symmetric instances on complete graphs involving up to 200 nodes, drawn from the TSPIB [9] library. The numerical results are reported in Table 2. Finally, we consider three asymmetric instances from the TSPIB [9] library that were also considered by Sherali and Driscoll [10]. Table 3 provides the linear relaxation values correspondingto the TOP TSP, Sherali and Driscoll SD and 3-TOP TSP formulations. These are compared with the optimal solution values OPT. As before, GAP denotes the dierence between the optimal value and the lower bound. References [1] A. Claus, A new formulation for the travellingsalesman problem, SIAM J. Algebraic Discrete Methods [2] G.B. Dantzig, D.R. Fulerson, S.M. Johnson, Solution of a large-scale traveling salesman problem, Oper. Res [3] M. Desrochers, G. aporte, Improvements and extensions to the Miller Tucer Zemlin subtour elimination constraints, Oper. Res. ett [4] P.J. Driscoll, A new hierarchy of relaxation for 0-1 mixed integer problems with application to some specially structure problems, Doctoral Dissertation, Department of Industrial and Systems Engineering, Virginia Tech, Blacsburg, Virginia, [5] M.M. Flood, The travelingsalesman problem, Oper. Res [6] M. abbe, P. Marcotte, G. Savard, A bilevel model of taxation and its application to optimal highway pricing, Management Sci [7] A. angevin, F. Soumis, J. Desrosiers, Classication of travellingsalesman problem formulations, Oper. Res. ett [8] E. awler, J.K. enstra, A.H.G. Rinnoy Kan, D.B. Shmoys Eds., The travelingsalesman problem. A Guided Tour of Combinatorial Optimization, Wiley, Toronto, [9] TSPIB, ibrary of travelingsalesman problems; www. iwr.uni-heidelberg.de/groups/comopt/software/ TSPIB95. [10] H.D. Sherali, P.J. Driscoll, On tightening the relaxations of Miller Tucer Zemlin formulations for asymmetric traveling salesman problem, Oper. Res [11] R.T. Wong, Integer programming formulation of the travelling salesman problem, Proc. IEEE Internat. Conf. Circuits Comput