The monotonic diameter of traveling salesman polytopes

Operations Research Letters 22 (1998) 69–73

The monotonic diameter of traveling salesman polytopes Fred J. Rispoli Department of Mathematics, Dowling College, Oakdale, NY 11769, USA Received 1 October 1996; revised 1 December 1997

Abstract The monotonic diameter of the polytopes arising in the asymmetric and the symmetric traveling salesman problem (TSP) are obtained. For the asymmetric TSP polytope associated with the complete directed graph on n nodes, the monotonic diameter is shown to be exactly bn=3c, for all n¿3. For the symmetric TSP polytope associated with the complete undirected graph on n nodes, the monotonic diameter is shown to be exactly bn=2c − 1, for all n¿3, except for n = 5, in which case it c 1998 Elsevier Science B.V. All rights reserved. is bn=2c. Keywords: Traveling salesman problem; Polytope; Diameter; Perfect matching

1. Introduction The diameter of a convex polytope P; (P), is the maximum distance between all pairs of extreme points of P. The “monotonic diameter” of P is de ned as follows. Let  be a linear form de ned on P and x any extreme point of P. Denote by (P; ; x) the number of edges of P in a shortest path of pairwise neighboring extreme points x = x0 ; x1 ; : : : ; x t = x∗ linking x to some extreme point x∗ minimizing , such that (x s )¿(x s+1 ). Let (P; ) = maximum{(P; ; x): x is an extreme point of P}: The monotonic diameter of P is ∗ (P) = maximum{(P; ): is a linear form de ned on P}: It is important to observe that for every convex polytope P, ∗ (P) exists and (P)6∗ (P). These facts will be used in the upcoming proofs. c 1998 Elsevier Science B.V. All rights reserved 0167-6377/98/$19.00 PII S 0 1 6 7 - 6 3 7 7 ( 9 8 ) 0 0 0 1 1 - X

Interest in the functions  and ∗ is motivated by simplex algorithms for linear programming (see [4–6]). Explicit formulas for  and ∗ have been established for various important network optimization problems including: the perfect matching and shortest path polytopes, the perfect 2-matching polytope, and a class of transportation polytopes that includes the assignment polytope as a special case (see [10, 11, 1]). Let Dn denote the complete directed graph on nodes N ={ 1; 2; : : : ; n} with edges {(i; j) ∈ N × N: i 6= j}, and let TSPn denote the polytope associated with the traveling salesman problem whose underlying network is Dn . Here we show that ∗ (TSPn ) = bn=3c, for all n¿3. Prior to this, the best known upper bound on ∗ (TSPn ) was given by the dimension of TSPn which is n2 −3n+1 (this follows from a result of Matsui and Tamura [7]), and no lower bound was established. It is known that (TSPn ) = 2, for all n¿6 (see Padberg and Rao [8]). Thus,  and ∗ are dierent for TSPn . Let Kn denote the complete undirected graph on nodes N with edges {{i; j}: i; j ∈ N; i 6= j}, and

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let STSPn denote the polytope associated with the symmetric traveling salesman problem whose underlying network is Kn . We also show here that ∗ (TSPn ) = bn=2c − 1, for all n¿3, except for n = 5, in which case it is bn=2c. In addition, a discussion is given expressing the above monotonic diameters in terms of the best possible upper bound for the diameter of polytopes arising in traveling salesman problems associated with subgraphs of complete graphs. The remainder of the paper is structured as follows. In Section 2 we consider certain faces of TSPn and use them to obtain ∗ (TSPn ). In Section 3, the symmetric traveling salesman polytope is discussed. The paper concludes with remarks related to the monotonic diameter of traveling salesman polytopes. 2. The monotonic diameter of TSPn A tour in Dn is a directed cycle of length n. The asymmetric traveling salesman polytope associated with Dn is de ned by TSPn = convex hull{xT: T is a tour in Dn }, where xT is the 0–1 incidence vec2 tor in Rn −n associated with T . The extreme points of TSPn are in one-to-one correspondence with the tours in Dn . For convenience, we refer to an extreme point xT in TSPn using its corresponding tour T . An assignment A in Dn is a subgraph consisting of n edges such that every vertex has in-degree and out-degree 1. An assignment is a set of disjoint directed cycles in Dn such that every node is on exactly one cycle. Thus, every tour is an assignment, but the converse is false since an assignment may contain “subtours”. The assignment polytope associated with Dn is de ned by APn = convex hull {x A: A is an assignment in Dn }, 2 where x A is the 0–1 incidence vector in Rn −n associated with A. Let K n; n denote the complete bipartite graph with nodes N ∪ N and edges N × N . Note that APn is dierent from the usual assignment polytope associated with K n; n , denoted by Pn; n , since the network Dn associated with APn has no edges of the form (i; i) (see [2] for more details on Pn; n ). The extreme points of APn are in one-to-one correspondence with the assignments in Dn and we refer to an extreme point x A in APn using its corresponding assignment A. APn provides an important tool for studying TSPn since TSPn ⊆ APn , and using matchings it is possible to obtain a “good” characterization of

neighbors on APn . A good characterization of neighboring extreme points on TSPn is unknown and it has been shown that the problem of determining if two arbitrary tours are not neighbors on TSPn is NP hard [9]. A subgraph M in K n; n is called a perfect matching if M consists of n edges and every node has degree 1 in M . Every assignment A in Dn corresponds to a perfect matching M in K n; n in an obvious way: (i; j) ∈ M i (i; j) ∈ A; and every perfect matching in K n; n without edges of the form (i; i) corresponds to an assignment. For any two sets S and T , let S
T denote the symmetric dierence (S ∼ T ) ∪ (T ∼ S) or equivalently (S ∪ T ) ∼ (S ∩ T ). In [2] it was shown that perfect matchings M1 and M2 in K n; n are neighbors on Pn; n if and only if M1
M2 contains a unique cycle. Clearly, this holds for APn as well. This gives the following Lemma. Lemma 1. Let A1 and A2 be assignments in Dn ; and let M1 and M2 be their corresponding perfect matchings in K n; n ; n¿3. Then A1 and A2 are neighbors on APn if and only if M1
M2 contains a unique cycle. In addition, the graph determined by the extreme points and extreme edges of APn is isomorphic to the graph of the face of Pn; n determined by the intersection of the facets of Pn; n associated with edges (i; i). Given any pair of assignments A1 and A2 in Dn , let FA (A1 ; A2 ) = convex hull{x A: A is an assignment in Dn satisfying A ⊂ (A1 ∪ A2 )}; and given any pair of tours T1 and T2 in Dn , let FT (T1 ; T2 ) = convex hull{xT: T is a tour in Dn satisfying T ⊂ (T1 ∪ T2 )}: Lemma 2. Let T1 and T2 be any pair of tours Dn ; and let M1 and M2 be their corresponding perfect matchings in Kn; n ; n¿5. If M1
M2 consists of p disjoint cycles, then: (a) FA (T1 ; T2 ) is a p-dimensional hypercube; and (b) FT (T1 ; T2 ) is a face of TSPn of dimension at most p. Proof. (a) Given any subset of q alternating cycles in M1
M2 ; 0¡q¡p, by exchanging M1 edges for M2 edges along these q cycles a perfect matching M in

F.J. Rispoli / Operations Research Letters 22 (1998) 69–73

M1 ∪ M2 is obtained. Since M1 ∪ M2 does not contain any edges of the form (i; i); M corresponds to an assignment A contained in T1 ∪ T2 . Conversely, from any perfect matching M in M1 ∪ M2 , we can use M1
M to identify a unique subset of cycles in M1
M2 . Thus, there is a one-to-one correspondence between the assignments A contained in T1 ∪ T2 and the subsets of {1; 2; : : : ; p}, implying that FA (T1 ; T2 ) contains 2p extreme points. In addition, given any pair of assignments A1 and A2 in T1 ∪ T2 , by Lemma 1, the distance between A1 and A2 on FA (T1 ; T2 ) is exactly the number of cycles in the symmetric dierence of the perfect matchings corresponding to A1 and A2 . Consequently, FA (T1 ; T2 ) must be a p-dimensional hypercube. (b) For n¿5; xij ¿0 de nes a facet of TSPn , for every edge (i; j) in Dn (see [3] for more details). Since FT (T1 ; T2 ) is the intersection of the facets {xij ¿0: (i; j) ∈= T1 ∪ T2 }; FT (T1 ; T2 ) is a face of TSPn , for every pair of tours T1 and T2 . Moreover, FT (T1 ; T2 ) ⊆ FA (T1 ; T2 ), which implies that the dimension of FT (T1 ; T2 ) is at most p. It is possible to construct examples for which this bound is obtained (one is given in the proof of Theorem 1.) Theorem 1. ∗ (TSPn ) = bn=3c, for every n¿3. Proof. First note that (TSPn ) = 1, for n = 3, 4 and 5, therefore ∗ (TSPn ) = bn=3c, for n = 3, 4, and 5. So assume that n¿5, and let  be a linear form de ned on TSPn . Let T ∗ be a tour on which  is minimized, and let T0 be any other tour satisfying ∗ (xT0 )¿(xT ). Since FT (T0 ; T ∗ ) is a face of TSPn which is itself a polytope, there exists an extreme point T1 on FT (T0 ; T ∗ ) such that (xT0 )¿(xT1 ) and T1 is a neighbor to T0 . T1 must also satisfy (T0 ∩ T ∗ ) ⊂ T1 ⊂ (T0 ∪ T ∗ ). Moreover, T0 6= T1 implies that T1 contains at least three edges that are not in T0 . Hence, T1 contains at least three more edges of T ∗ than T0 . So, after at most bn=3c repetitions of this observation, we can obtain the tour T ∗ . Consequently, ∗ (TSPn )6bn=3c for n¿5. To demonstrate that ∗ (TSPn ) = bn=3c, rst note that (TSPn ) = 2, for n¿5, implies ∗ (TSPn ) = bn=3c, for n = 6, 7 and 8. The following example shows that the bound is obtainable for every n¿9. Let k¿3, and set: T1 = (1; 2; 3; : : : ; n; 1);

for all n¿9;

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T2 = (1; 3; 2; 4; 6; 5; 7; : : : ; n − 2; n; n − 1; 1); for all n = 3k; T2 = (1; 3; 2; 4; 6; 5; 7; : : : ; n − 3; n − 1; n − 2; n; 1); for all n = 3k + 1; and T2 = (1; 3; 2; 4; 6; 5; 7; : : : ; n − 4; n − 2; n − 3; n − 1; n; 1); for all n = 3k + 2: Let M1 and M2 be the perfect matchings in K n; n corresponding to T1 and T2 (there are three variations of M2 depending on whether n = 3k; 3k +1, or 3k +2, but we may treat these cases identically). In every case M1
M2 consists of bn=3c disjoint cycles. It follows from Lemma 2 that FA (T1 ; T2 ) is an bn=3c-dimensional hypercube. However, every assignment A contained in T1 ∪ T2 is also a tour. Therefore, every extreme point on FA (T1 ; T2 ) is also an extreme point on FT (T1 ; T2 ). Hence, FT (T1 ; T2 ) = FA (T1 ; T2 ). Now let (x) = cx be the objective function given by:  0 if (i; j) ∈ T2 ;     if (i; j) ∈ T1 − T2 and; Cij = 1     n + 1 if (i; j) ∈ Dn − (T1 ∪ T2 ): Since  attains its minimum only at T2 ; (xT2 ) = 0; (xT1 )6n, and (xT )¿n, for every T in Dn − (T1 ∪ T2 ), it follows that every path nonincreasing in  linking T1 to T2 on TSPn must be on the face FT (T1 ; T2 ). Moreover, FT (T1 ; T2 ) = FA (T1 ; T2 ) and Lemma 1 imply that two tours are neighbors on FT (T1 ; T2 ) if and only if their corresponding perfect matchings have a symmetric dierence containing a unique cycle. Hence, no single step can obtain edges from more than one cycle in M1
M2 . Since M1
M2 contains bn=3c cycles, at least bn=3c steps are needed to join T1 to T2 by a path nonincreasing in  on FT (T1 ; T2 ). Therefore, ∗ (TSPn ; )¿bn=3c, which implies that ∗ (TSPn ) = bn=3c, for all n¿9.

3. The symmetric traveling salesman polytope A tour in Kn is a cycle of length n. De ne the symmetric traveling salesman polytope associated with Kn to be STSPn = convex hull {xT: T is a tour in Kn },

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where xT is the 0–1 incidence vector in Rn(n−1)=2 associated with T . Now the “perfect 2-matching polytope” can be used to study STSPn . A perfect 2-matching is a subgraph of Kn consisting of n edges such that every node in M has degree exactly 2. The perfect 2-matching polytope associated with Kn is de ned by PTMn = convex hull of {xT : T is a perfect 2− matching in Kn }. In [11] it was shown that perfect 2-matchings M1 and M2 are adjacent on PTMn if and only if M1
M2 contains a unique alternating cycle. Since two tours that are neighbors on PTMn must also be neighbors on STSPn we have the following. Lemma 3. Let T1 and T2 be tours in Kn ; n¿4. If T1
T2 contains exactly one alternating cycle, then T1 and T2 are neighbors on STSPn . In [12] Lemma 3 is used to obtain a bound on (STSPn ). Now, the proof of Theorem 1 is extended to the symmetric case. One can check directly that ∗ (STSP3 ) = 0; ∗ (STSP4 ) = 1, and ∗ (STSP5 ) = 2. So let n¿5, and suppose that T1 and T2 are tours in Kn . It still holds that the convex hull of {xT: T is a tour in Kn satisfying T ⊂ (T1 ∪ T2 )} is a face of STSPn . Let  be a linear form de ned on STSPn ; T ∗ a tour on which  is minimized, and T0 any other tour satisfying ∗ (xT0 )¿(xT ). As in the proof of Theorem 1, we are guaranteed the existence of T1 which is a neighbor to T0 , however, now T1 must contain at least two edges of T ∗ instead of three. So repeating this observation at most b(n − 4)=2c times gives a tour Tk satisfying |Tk ∩ T ∗ |¿n − 4. It follows from (STSP4 ) = 1, that two tours in Kn intersecting in n − 4 or more edges must be neighbors on STSPn , for all n¿5. Hence, Tk and T ∗ are neighbors, and ∗ (STSPn )6bn=2c − 1, for n¿5. Using Lemma 3, one can deduce that (STSPn ) = 2, for n = 6 and 7, details are left for the reader. This fact, together with the above discussion, gives ∗ (STSPn ) = 2, for n = 6 and 7. It was shown in [13] that faces of STSPn exist that are (bn=2c − 1)dimensional hypercubes, for n¿8. Therefore, a linear objective function  de ned on STSPn exists such that (bn=2c − 1) steps are required to link two tours with a path that is nonincreasing in . This proves Theorem 2.

Theorem 2. ∗ (STSPn ) = bn=2c − 1; for every n¿3; except for n = 5; in which case ∗ (STSP5 ) = 2: 4. Concluding remarks In this paper the monotonic diameters of the asymmetric and the symmetric traveling salesman polytopes are obtained. These results can be interpreted to yield the best possible upper bound on the diameter of a traveling salesman polytope associated with sparse underlying networks. That is, given any subgraph G of Dn containing at least two directed tours, let TSPn (G) denote the convex hull of the incidence vectors of all of the directed tours in G. Then the proof of Theorem 1 shows that (TSPn (G))6bn=3c holds for all G. The bound is the best possible since there exists a subgraph G for which (TSPn (G)) = bn=3c. Namely, take G = T1 ∪ T2 , where T1 and T2 are de ned as in the last part of the proof of Theorem 1. This clearly demonstrates that the result (TSPn ) = 2; n¿6, depends on the underlying network being a complete graph. When this assumption is removed, the best possible upper bound on the diameter of the asymmetric traveling salesman polytope associated to arbitrary subgraphs of Dn is bn=3c. In the symmetric case we have the following. Given any subgraph G of Kn containing at least two (undirected) tours, let STSPn (G) denote the convex hull of incidence vectors of tours in G. Then (STSPn (G))6bn=2c − 1, for all n¿6. Finally, we note that this paper gives the rst explicit monotonic diameter for an NP-complete problem. Interestingly, the computational diculty in solving the traveling salesman problem is not re ected in the monotonic diameter. For example, the monotonic diameter of the symmetric traveling salesman polytope associated with Kn is roughly the same as that of the perfect 2-matching polytope associated with Kn , which is bn=2c. Yet strongly polynomial algorithms are known to exist for the perfect 2-matching problem, but no strongly polynomial algorithm is known to exist for the symmetric traveling salesman problem.

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