A constructive method is presented for optimizing exactly the Traveling Salesman Problem as a sequence of shortest route problems. The method combines group theoretic and Lagrangean relaxation constructions. Key Words: Traveling Salesman Problem, Lagrangean relaxation, shortest route problem, generalized linear programming, group theory

Shapiro, Jeremy F., 1939-

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''- ' --'..'''MASScHSETTS IN,,TT T .T· .. . . . . . . ..... :~~~ ~ ~ ~--.:·.rii· ··-..ir 1-.: :--' · ·- -'.-I··. -- .'--·. · ·. ·-;.. i::·:`··...i.i: . : ;i·.· ·- · .- - ·.. .I·. ·.- ·- ·:·i .;·le· ·. ·. ri.::-.-.;.--. -..r.-i . I· :.·.-- . .:.-·:Convergent Duality for the TravelingSalesman ProblembyJeremy F. ShapiroOR 204-89 November 1989CONVERGENT DUALITY FOR THETRAVELING SALESMAN PROBLEMJeremy F. ShapiroOperations Research CenterMITAbstractA constructive method is presented for optimizing exactly the TravelingSalesman Problem as a sequence of shortest route problems. The method combinesgroup theoretic and Lagrangean relaxation constructions.Key Words: Traveling Salesman Problem, Lagrangean relaxation, shortest routeproblem, generalized linear programming, group theory.1. IntroductionIn this paper, we present a constructive method for optimizing exactly thetraveling salesman problem (TSP) by solving a sequence of shortest route problems.The method is a synthesis of two computational methods: (1) Lagrangeanrelaxation methods for formulating the TSP as a shortest route problem with sideconstraints (Picard and Queyranne [6], Houch et al [4], Fox et al [3]); and (2) grouptheoretic methods for resolving duality gaps in integer programs (Bell and Shapiro[1]). The two constructive procedures merge seamlessly because the groupoptimization problems required to resolve duality gaps are themselves shortestroute problems.The plan of this paper is as follows. Section 2 contains a statement of the TSPas a shortest route problem with side constraints and the ordinary relaxation of thisproblem. In Section 3, we discuss how the ordinary Lagrangean can be iterativelystrengthened until some shortest route calculation yields a TSP tour, which byconstruction must be an optimal tour. A numerical example is presented inSection 4. A few concluding remarks are given in Section 5.12. Statement of the Model and the Initial DualConsider a network with nodes 0, 1, 2, ..., N, directed arcs (i, j) for all i and j i,and associated arc lengths cij. The arc lengths need not be symmetric; that is, weallow'cij * cij. We define a new node with the label N+1 which is located identicallywith node 0 but corresponds to terminating there. Thus, the arc costs ci,N+l = CiO forall i. The TSP can be expressed as the problem of sending one unit of flow fromnode 0 to node N+1 so that the distance traveled is minimized (this is a shortestroute problem), but with the additional constraints that each city must be visitedexactly once. Shortest route solutions that satisfy the additional constraints arecalled tours. The construction is illustrated in Figure 1 (without all possible arcs) fora 7 node problem.Shortest Route Representation of TSPFigure 1Letting xijt (ij) denote zero-one decision variables, the model can be stated asTraveling Salesman ProblemN N1- N N NV = min Z cj xojo + Z , cij xijt + Y Ci,N+l xi,N+I,Nj=l t=l i=l j=l i=lji(1)2Subject toNx0j = 1j=lN-XOiO + Ij=lj~sxijl = 0NXki,t-1 + xijt = 0j=1joiXki,N- 1 + Xi,N+1,N = 0for i=l,...,Nt=2,...,N-1for i=l,...,NXi,N+l,N = 1Ni=lN-1 NXjo + Xijt = 1t=l i=liejfor j=l,...,Nxijt = 0 or 1The constraints (2)- (4) describe the shortest route problem. The constraints (5)require that each node be visited exactly once.The constraints (5) are the ones that make the model combinatoriallycomplex. The traditional Lagrangean relaxation approach in such a circumstance isto dualize on these constraints. Letting rj denote the dual variable on row j in (5),and letting ;i denote the N-vector of these variables, the result is3(2)for i=l,...,NN-Zk=lksiNk-1k*i(3)(4)(5)(6)N N N-1 N N NLO(+) m= (-j + + (cij - j)xijt + Z Ci,N+l Xi,N+l,N (7)j=1 j=1 t=l i=1 j=l i=ljisubject to: (2), (3), (4), (6)The Lagrangean calculation (7) is a shortest route problem in which the arc lengths cjare adjusted to cij - j to reflect the relative economy or diseconomy of visitingnode j.Let xij t () denote an optimal solution to (6). It is easy to show that thissolution is optimal in TSP ifN-1 NXo0j () + Z E Xijt () = 1 for j=l,...,N (8)t=l i=l (8)izjThus, if the Langrangean LO, or any of the strengthened Lagrangeans derived in thenext section, ever yields an optimal shortest route path that is a TSP tour, such atour is optimal. To try to select a n E RN that will cause this fortuitous outcome, weare led to consider the dual problemTSPD(O)DO = max LO() (9)s.t. e RNSince LO () < V for all 7t e RN ,we have DO < V. It is also easy to show that if 7tproduces an optimal solution to TSP, n is optimal in the dual TSP (9); moreover,DO = V. Thus, the indicated constructive approach is first to compute r* that isoptimal in TSPD(O), and then seek xijt (X*) that satisfies the degree constraints (8).The deficiency of the Lagrangean approach described thus far is that wecannot guarantee the existence of any i such that xijt () satisfies (8). This deficiencyis equivalent to the condition that DO < V, in which case we say there is a dualitygap. The main purpose of this paper is to present a constructive method for4resolving duality gaps, thereby allowing the Lagrangean shortest route calculationsto continue until a solution satisfying (8) is computed.The point of departure for the new constructive method is the generalizedlinear programming algorithm (see [5] or [7]) for solving (9). LetXr = (Xijtr) for r=l,...,Roan enumeration of the xijt satisfying (2), (3), (4), (6). These correspond tofeasible paths in the shortest route network. LetN N-1Cr = Coj XOjOr + j=1 t=lNN NI Cij Xijtr + I Ci,N+l Xi,N+l,N,ri=I j=l i=1jfiN-1 Ndjr = XjOr+ I Xijtrt=l i=linjfor j=I,...,NIt is well known that TSPD(O) is equivalent to (see [5])MLP (0)DO = minRo2 Cr Orr-lfor j=l,...,NRos.t. djrfr=l r-lRor-1fr > 0This master LP has an enormous number of columns. Starting with a small subset,new columns can be generated iteratively by computing LO (i) where X is the vector5representthe set ofand(10)(11)(12)of optimal LP shadow prices on the degree rows in (12). This computation willeither establish that X is optimal in TSPD(O), or uncover a new column to add to theMaster (12). Generalized linear programming will compute an optimal x aftergenerating a finite number of new columns since there are only a finite number ofpaths in the shortest route network. Finally, we note that MLP (0) is equivalent tothe LP model that results if we relax the zero-one integrality restriction (6) and allowthe xijt to take on continuous values between 0 and 1. In particular, any feasiblesolution to MLP (0) correspondsRoxijt = Z Xijtr Orr=1which in turn is a feasible solution to the ordinary LP relaxation.Our interest in this paper is how to proceed when optimizing MLP(0) andtherefore TSPD(0) fails to find an optimal solution to TSP. This is the subject of thefollowing section.3. Iterative Dual Model ConstructionWe begin with a brief discussion, which is merely algebraic and non-constructive, about how the Lagrangean L0 () can be strengthened if it fails to yieldan optimal solution to TSP. The remainder of this section is devoted toconstructive procedures for selecting the algebraic groups from which the strongerLagrangeans are derived. The reader is referred to Shapiro [71 for background.The degree constraints (5) are equations defined over ZN, the infinite abeliangroup of integer N-vectors under ordinary addition. These equations can beaggregated or relaxed as follows. Choose any finite abelian group G and anhomomorphism IV mapping ZN onto G. LetEij-(ej) for j=l, ... , Nwhere ej is the j-th unit vector in ZN and "E" denotes equality in G. The aggregationof (5) is achieved by multiplying each equation by ej and summing. The result isN N N- NI ejx ojo+ + I I ejXit E (13)j= j=-1 t=l i=liwjwhere6N_ £ E.j=lIt is straightforward to demonstrate that any zero-one solution xijt that satisfies (5)satisfies (13)The new Lagrangean isN N N-IN N NL17) = Atj + min A (coj - )x0jo + , Y I (cij - n j) Xijt + , CN+ xi,N+l,N (14)j=1 j=l t=l i=l j=l i=ljfisubject to: (2), (3), (4), (13), (6)In particular, the Lagrangean has been strengthened in the sense that LV () > LO (7C)for any e R N because we have added the constraint (13). The Lagrangean Lv is still ashortest route problem, but over an expanded network. Formally, the nodes in thenetwork comprise the starting and ending nodes 0 and N+1 as before, plus nodeslabelled i, t, a for i=1,...,N; t=l,...,N; and for all ao e G. The arcs in this network are:(i) (0, (j, 1, ej)) with length coj - 7tj for j=1,...,N(ii) ((i, t, a), (j, t+l, a+Ej)) with length cij - tj for i=l,...,N:j=l,...,N and joit=l,...,N-1and(iii) ((i, N,P), N+1) with length Ci,N+l for i=l,...,NNote that the terminal node N+1 can be reached only from nodes (i, N,P) for some i.An example of the expanded network is given in Figure 3.We turn now to constructive procedures that rationalize the choice of G andV. The construction begins with the assumption that the ordinary Lagrangeanrelaxation L () and associated dual problem TSPD(O), which we optimize bygeneralized linear programming and the simplex method, fails to find an optimalsolution to the original TSP. The construction is based on the observation that7MLP(O) with the additional constraint that the variables Or = 0 or 1 solves TSP.With this constraint and all feasible columns included, the model would select ashortest route path that visits nodes 1,2,...,N exactly once before returning to node 0(that is, node N+1).Thus, viewing MLP(O) as the linear programming relaxation of the integerprogram that we wish to solve, we apply general purpose integer programmingmethods based on number (abelian group) theory to the relaxation if it fails toproduce an optimal solution to TSP. The following theorem characterizes thepossible outcomes of the dual analysis of the previous section.Theorem 1: Suppose MLP(O) is optimized using the simplex method. Let Or forsome r=l,...,R denote an optimal solution, where R • Ro is the number of columnsgenerated by generalized linear programming in solving TSPD(O).Case (a): If Ss = 1 for some s, (xijts) is an optimal TSP tour.Case (b): If more than one 2r is positive, say fr > 0 for r = 1,2,...,K with K 2 2, (xijtr)is not a feasible TSP tour for r = 1,2,...,K.Proof: The proof is the same as Lemma 2 of [1].The ordinary Lagrangean relaxation and analysis of the previous section failsto solve TSP under case (b) of Theorem 1. This probably means that Do < V and weneed to somehow strengthen the Lagrangean to close up the gap. It may be,however, that DO = V but we have not yet identified a TSP tour with length DO viathe linear program MLP(O) and the implied column generation procedure. Theconstructive procedure we are about to present will in any event resolve thedifficulty.Suppose Case (b) of Theorem 1 obtains. The method proceeds by consideringthe (N+l)xK matrixd 1 d 12 [dKQ j d 18This integer matrix induces a unique abelian group G1 and a homomorphism MVlmapping ZN+1, the infinite abelian group of integer (N+1) vectors under ordinaryaddition, onto G1. The mapping can be characterized by computingV1 (e j)- j for j=I,...,N+lsince for any a E ZN+lN+1Vl(a)=- ajejj=1The mapping has the property thatNdr £j + N+l - 0 for j=I,...,Kj=1whereasN+1Ej + N+l Oj=1since the computed optimal solution for MLP(O) is not integer. LettingN3 = e £j, we havej=1NX djr£ji 3 for j=l,...,K (15)j=1The group G1 and the mapping can be computed by using the Smith reductionprocedure (see [7]) to diagonalize Q.We use V1 in strengthening the Lagrangean to the form (14), thereby creatinga stronger dual problem. By (15), the new Lagrangean is strictly stronger than L inthe sense that each shortest route path (xijtr) corresponding to an infeasible tour thatwas given a positive weight r > 0 in MLP(O) is infeasible in the Lagrangean LY.9There is no guarantee, however, that the dual problem TSPD(1) analogous to(9) defined for L'1 will yield an optimal TSP tour. In particular, we may find thatcase (b) of Theorem 1 obtains for the Master linear program MLP(1) that solvesTSPD(1). In such an event, we repeat the construction just outlined to derive agroup G 2, a homomorphism 2, and a still stronger Lagrangean L2. Equation (13) isnow written over the abelian group G1 @ G2. If necessary, the procedure can beapplied an arbitrary number of times. Since we eliminate at least two shortest routepaths each time we strengthen the Lagrangean and since the number of such pathsis finite, our constructive procedure must terminate finitely with a Lagrangean, avector e RN, and a solution xijt () such that (8) holds; that is, such that xijt (n) is anoptimal tour. Thus, we have proved the following.Theorem 2: The constructive dual procedure for the TSP will, in a finite number ofincreasingly strong dual model constructions, produce a shortest route network anda vector of dual variables X e RN such that the calculated shortest route path(Lagrangean optimization) will be an optimal TSP tour.4. Numerical ExampleWe illustrate the procedure for a 7 city TSP with the distance matrix shown inTable 1.ToCityFrom 1 2 3 4 5 6City0I234552 80 120 71 218 17857 116 90 188 13961 61 121 10158 132 115176 1546210The initial dual problem TSPD(O) produces two optimal shortest routes paths,as depicted in Figure 2, that correspond to optimal columns in MLP(O), each withweight equal to 1/2. These columns induce the abelian group G1 = (0, 1) and thehomomorphism iWl with the property N1 (el) - 1 (e2) - i1 (e6) - 0 andNll (e3)0V1 (e4)- 1 (e)_ 1.The extended shortest route network using G1 and N1 is shown in Figure 3.The solid lines represent the shortest route tours of Figure 2 except for the finaltransition to node 7 which is not allowed since these paths are drawn to nodes att = 6 on the replicate a = O0. By contrast, the optimal TSP tour (0, 1, 2, 6, 5, 3, 4, 7) isdepicted with a dotted line. The transition to node 7 from the replicate a = 1 isallowed according to our choice of G1 and 1 .Shortest Route Paths Producing a Duality GapFigure 211IIIIIIIIIC IC 20IIo10 @Xo b'@I'i'1 ': l W.I0o1I II Ia II II I I II II S00 M Q,Ia I=" :0 0 000a =1Extended Shortest Route Representation of TSP· Figure 312;,rOm__, S s -S - l5. ConclusionsThe constructive method presented in this paper computes an optimal TSPtour by solving a finite sequence of shortest route problems. Small computationalexperiments performed to date with the method are insufficient for judging theultimate effectiveness of the method. The experiments do indicate, however, aninteresting array of constructive ambiguities whose study might reveal new insightsinto the TSP. For example, the underlying group structures change radically if thedegree constraints (8) are written asN-1 NXOjO ()+ I Xijt () 1 for j=1,...,Nt=1 i=liwjThese inequalities will, of course, be satisfied as equalities by any TSP tour.Another ambiguity is the selection of columns from the linear programmingbasis in the generalized linear programming Master from which to derive the groupand associated homomorphism. The TSP has been optimized exactly if any column,even one with zero weight (that is, a degenerate column), corresponds to a TSP tour.Otherwise, all columns in the basis with weight equal to zero can be included in thematrix form from which the group and the homomorphism are derived since theyare not tours and might as well be eliminated in the strengthened Lagrangean.The method presented here was motivated in part by column generationapplications involving the TSP. One such application to vehicle routing andscheduling is given in [2]. As we have shown, integer programming columngeneration involving TSP's can be made theoretically exact by our method. Thepractical use of the method awaits further experimentation.6. References[1] D. E. Bell and J. F. Shapiro, "A convergent duality theory for integerprogramming," OQns. Res., 25, pp 419-434 (1977).[2] D. Cattrysse and J. F. Shapiro, "Column generation models for vehicle routingand scheduling problems," submitted for publication.[3] K. R. Fox, B. Gavish, and S. C. Graves, "An n-constraint formulation of the(time-dependent) traveling salesman problem," Opns. Res., 28, pp. 1018-1021(1980).13_ II_ --· ·---- ---[4] D. J. Houck, Jr., J-C. Picard, M. Queyranne, and R. R. Vemuganti, "Thetraveling salesman problem as a constrained shortest path problem: Theoryand computational experience," Opsearch, 17, pp 93-109 (1980).[51 T. L. Magnanti, J. F. Shapiro and M. H Wagner, "Generalized linearprogramming solves the dual," Man. Sci., 22, pp 1195-1203, (1976).[6] J-C. Picard and M. Queyranne, "The time-dependent traveling salesmanproblem and applications to the tardiness problem in one-machinescheduling," Opns. Res., 26, pp 86-110 (1978).17] J. F. Shapiro, Mathematical Programming: Structures and Algorithms,Wiley-Interscience (1979).14

Convergent Duality for the Traveling Salesman Problem

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Convergent Duality for the Traveling Salesman Problem

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