D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Info. Proc. Lett., 35(6):281-285, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....optimal. An approximation algorithm is called k optimal if the the feasible solution it generates is within a factor k of the optimal solution. Currently the best known results for the TSP are a 3 2 approximation algorithm for instances when the edge weights satisfy the triangular inequality [5, 15], and an (1 ) approximation algorithm for the Euclidean TSP [3] It is also known that, unless P=NP, there cannot be a (1 ) approximation algorithm for general TSP. Initially approximation algorithms relied on suitably rounding the solutions of linear programming relaxations of the ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Info. Proc. Lett., 35(6):281-285, 1990.

....Two decades of research failed to improve upon Christofides algorithm for metric TSP. The Held Karp heuristic is conjectured to have an approximation ratio 4 3 (some results of Goemans [27] support this conjecture) but the best upperbound known is 3 2 (Wolsey [63] and Shmoys and Williamson [58]) Some researchers continued to hope that even a PTAS might exist. A PTAS or Polynomial Time Approximation Scheme is a polynomial time algorithm or a family of such algorithms that, for each fixed c 1, can approximate the problem within a factor 1 1 c. The running time could depend ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: A monotonicity property wtih application. Information Processing Letters, 35:281--285, 1990.

....cy = 2cx = 2 minfcx j x 2 ToC(G)g: The rst equality here follows from the de nition of B. The second equality is equation (4) and the inequality is true because y is feasible for A. The nal two equalities follow from the de nitions of y and x . ut Wolsey [11] and Shmoys and Williamson [9] prove the following theorem. Theorem 3. Let G = V; E) be a graph with edge weight function c satisfying the triangle inequality. Then the weight of the traveling salesman tour on G output by Christo des algorithm is no more than 3 2 minfcx j x 2 ST (G)g. From Theorems 2 and 3, and the fact ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Information Processing Letters, 35:281{ 285, 1990.

....1.3 Related work A heuristic for finding a low cost Hamilton cycle was developed by Christofides in 1976 [4] An analysis of this heuristic shows that the ratio is no worse than 3 2 in both Conjecture 1 and Conjecture 2. This analysis was done by Wolsey in [16] and by Shmoys and Williamson in [15]. A modification of the Christofides heuristic to find a low cost 2 vertex connected subgraph when the costs obey the triangle inequality was done by Fredrickson and Ja Ja in [5] The performance guarantee for this heuristic to find a 2 vertex connected subgraph is 3 2 . There has also been a ....

D.B. Shmoys and D.P. Williamson, Analyzing the Held-Karp TSP bound: A monotonicity property with application, Inf. Process. Lett. 35 (1990) 281-285.

....Two decades of research failed to improve upon Christofides algorithm for metric TSP. The Held Karp heuristic is conjectured to have an approximation ratio 4 3 (some results of Goemans [26] support this conjecture) but the best upperbound known is 3 2 (Wolsey [62] and Shmoys and Williamson [57]) Some researchers continued to hope that even a PTAS might exist. A PTAS or Polynomial Time Approximation Scheme is a polynomial time algorithm or a family of such algorithms that, for each fixed c 1, can approximate the problem within a factor 1 1 c. The running time could depend upon ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: A monotonicity property wtih application. Information Processing Letters, 35:281--285, 1990.

.... cy = 2cx = 2 minfcs j x 2 ToC(G)g: The rst equality here follows from the de nition of B. The second equality is equation (4) and the inequality is true because y is feasible for A. The nal two equalities follow from the de nitions of y and x . Wolsey [11] and Shmoys and Williamson [9] prove the following theorem. Theorem 2.3. Let G = V; E) be a graph with edge weight function c satisfying the triangle inequality. Then the weight of the traveling salesman tour on G output by Christo des algorithm is no more than 3 2 minfcx j x 2 ST (G)g. From Theorems 2.2 and 2.3, and the ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Information Processing Letters, 35:281-285, 1990.

....function c (HK) A laminar solution for the dual of the Held Karp relaxation of a Euclidean TSP has a nice geometric interpretation as a moat packing (see e.g. 38] We obtain an ffl approximately fair allocation from this using the following results: Theorem 3.4. 3 (Wolsey[61] Shmoys, Williamson[52]) If the distances of a Euclidean TSP satisfy the triangle inequality then the optimum value of (HK) is at least 2 3 of the length of a shortest tour. It is a well known open conjecture that the factor of 2 3 can be replaced by 3 4. Clearly, this result implies that the multiplicative 1 2 ....

....2 3 of the length of a shortest tour. It is a well known open conjecture that the factor of 2 3 can be replaced by 3 4. Clearly, this result implies that the multiplicative 1 2 core is non empty. Actually, the result of Shmoys and Williamson is even stronger. Theorem 3.4. 4 (Shmoys, Williamson [52]) If the distances of a Euclidean TSP satisfy the triangle inequality then the optimum value of (HK) is at least 2 3 of the length of a tour obtained by the method of Christofides (see e.g. 48] 48 Computational Aspects of Combinatorial Cooperative Games Thus, the cost c Chr of a tour ....

SHMOYS, D., AND WILLIAMSON, D. Analyzing the held-karp tsp bound: a monotonicity property with applications. Information Processing Letters 35 (1990), 281--285.

....determine an optimal moat packing in polynomial time. Since the feasible region Q n of the Traveling Salesman Polytope is contained in S n , any optimal solution to minimizing over S n is a lower bound for the optimal value of TSP. It was proved by Wolsey [32] and by Shmoys and Williamson [27] that for any distance function d satisfying the triangle inequality, this bound can be at worst 2 3 of the optimum: Theorem 3.5 ( 32] 27] If the distances satisfy the triangle inequality then the optimum value of (HK) is at least 2 3 of the length of a shortest tour. Euclidean TSP Games ....

....n , any optimal solution to minimizing over S n is a lower bound for the optimal value of TSP. It was proved by Wolsey [32] and by Shmoys and Williamson [27] that for any distance function d satisfying the triangle inequality, this bound can be at worst 2 3 of the optimum: Theorem 3. 5 ( 32] [27]) If the distances satisfy the triangle inequality then the optimum value of (HK) is at least 2 3 of the length of a shortest tour. Euclidean TSP Games 12 It is a well known open conjecture that the factor of 2 3 can be replaced by 3 4. This is known as the Held Karp Conjecture. It should be ....

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D.Shmoys, D.Williamson, Analyzing the Held-Karp TSP bound: a monotonicity property with applications, Information Processing Letters, 35 (1990), 281--285.

....graph CG resulted, and in any metric space 3 2 (T ST ) CG) In DocNumber 5 . 1. 7. Rao Smith typeset 793 May 23, 1998 Approximation schemes fact 3 2 (LP ) CG) where (LP ) is the cost of a simple linear programming lower bound on the TST cost (also computable in polynomial time) [37]. This approximation ratio 1:5 is still best known for TST in arbitrary metric spaces for approximation algorithms running in polynomial time. Smith [39] showed how, by searching over separators, to find the optimum TST for N points in the Euclidean plane in N O( p N) steps. Other ....

....in some heuristic manner. It seems impossible to tell how much effect these ploys will have, except by experiment. A brief survey of the competition is below. TST: In practice heuristic algorithms such as the LinKernighan [29] 34] 14] local optimization procedure and Held Karp lower bound [22] [37] will rapidly find a TST and a proof that it is within a small factor (usually a few percent) of optimality. No proof is known that the running time is always going to be rapid or that the approximation is always going to be good, but in practice it almost always is. Furthermore, programs that ....

D.B. Shmoys and D.P. Williamson. Analyzing the Held-Karp TSP bound: A monotonicity property with application. Info. Proc. Lett., pages 281--285, 1990.

....over the subtour polytope, obtained by adding the degree constraints x(ffifig) 2 for all i 2 V to SP . As a result, the value obtained by optimizing over SP is exactly equal to the Held Karp lower bound [18] Interpreting their result in terms of GTSP, Wolsey [31] and Shmoys and Williamson [30] have shown that Minfcx : x 2 GTSPg Minfcx : x 2 SPg 3 2 ; for any nonnegative cost function c. However, this bound does not appear to be tight and, in fact, the following conjecture motivated this study (see Sections 3 and 5) Conjecture 1 For any nonnegative cost function c, Minfcx : x 2 ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: A monotonicity property with application. Information Processing Letters, 35:281--285, 1990.

.... of all the odd valent points in the spanning tree, an Eulerian (even valent) graph CG resulted, and in any metric space 3 2 (T ST ) CG) In fact 3 2 (LP ) CG) where (LP ) is the cost of a simple linear programming lower bound on the TST cost (also computable in polynomial time) [22]. This approximation ratio 1:5 is still best known for TST in arbitrary metric spaces for approximation algorithms running in polynomial time. Smith [23] showed how, by searching over separators, to find the optimum TST for N points in the Euclidean plane in N O( p N) steps. Other ....

....in some heuristic manner. It seems impossible to tell how much effect these ploys will have, except by experiment. A brief survey of the competition is below. TST: In practice heuristic algorithms such as the LinKernighan [19] 21] 9] local optimization procedure and HeldKarp lower bound [14] [22] will rapidly find a TST and a proof that it is within a small factor (usually a few percent) of optimality. No proof is known that the running time is always going to be rapid or that the approximation is always going to be good, but in practice it almost always is. Furthermore, programs that ....

D.B. Shmoys and D.P. Williamson. Analyzing the HeldKarp TSP bound: A monotonicity property with application. Info. Proc. Lett., pages 281--285, 1990.

.... instances, asymptotically the bound is only about 0:7 different from the optimal tour length, and for the real world instances of TSPLIB [12] the gap is usually less than 2 [7] And if the distances obey the triangle inequality, the bound will be at least 2=3 of the length of the optimal tour [13, 15]. It is possible to give more complicated IPs whose relaxations have smaller gaps, but we did not attempt to work with them for reasons that we will explain after we have reviewed the method in more detail. Observe that it is not trivial to plug this linear program into an LP solver, because there ....

D. B. Shmoys and D. P. Williamson. Analyzing the held-karp tsp bound: A monotonicity property with applications. Inform. Process. Lett., 35:281--285, 1990.

.... U appears in at most ae covering sets (that is each errand can be completed at at most ae locations) Using again the optimal solution to the LP relaxation of the general ERRAND SCHEDULING problem (or TCP) plus the techniques developed for solving the PRIZE COLLECTING TSP (see for example [6] or [31]) we show that the ESP problem can be approximated within 3ae=2 and the TCP problem within ae. These contain as special cases the well known result of Christofides [9] for approximating TSP and the result of Hochbaum [20] where she proved that the SET COVER problem can be approximated to within ....

....any subset S of V , denote by L(S) the length of the optimal traveling salesman tour through vertices in S and by L C (S) the length of the tour passing through all vertices of S obtained by the well known Christofides algorithm. That is L(V ) Z 2 . Wolsey in [32] and Shmoys and Williamson in [31] proved the following: Proposition 4 L(V ) L C (V ) 3 2 Z 2 2 Let V 2 ae V . Define r i = ae 2 if i 2 V 2 , 0 otherwise. Consider the following LP problem. Problem LP 3 : Z 3 = minimize X e2E c e x e ; subject to (a) X e2ffi(fig) x e = r i ; for all i 2 V ; b) X ....

D.B. Shmoys and D.P. Williamson. Analyzing the Held-Karp TSP bound: A Monotonicity Property with Application. Information Processing Letters 35(1990), pp. 281-285.

....i j is maximized. Then branch on the inclusion exclusion of fi; jg: E [ ffi; jgg; I) E; I [ ffi; jgg) 7. MISC 117 Note that for the current 1 tree is an optimal solution to the second subproblem (but this subproblem has a bigger set of included edges) 7. Misc Shmoys and Williamson [40] show that the Christofides approximation is actually less than or equal to 1.5 times the Held Karp lower bound, as depicted in Figure XVIII.50. optimal solution Christofides upper bound Held Karp lower bound = 1.5 Held Karp lower bound Figure XVIII.50. Christofides upper bound = 1.5 Theta ....

D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP Bound: a Monotonicity Property with Application. Inf. Proc. Let., 35:281--285, 1991.

....of the TSP heuristic for finding a low cost Hamilton cycle developed by Christofides in 1976, see [2] As a consequence, it was shown that the approximation ratio is no worse than 3 2 in both Conjecture 3 and Conjecture 2. This analysis was done by Wolsey in [12] and by Shmoys and Williamson in [11]. A modification of the Christofides heuristic to finds a low cost 2 vertex connected subgraph was done by Fredrickson and Ja Ja in [3] The performance ratio for this heuristic to find a 2 vertex connected subgraph is 3 2 . There has also been a spate of work on approximation algorithms for the ....

D.B. Shmoys and D.P. Williamson, Analyzing the Held-Karp TSP bound: A monotonicity property with application, Inf. Process. Lett. 35 (1990) 281-285.

....value of the objective function is unchanged if we include the constraints x(ffi(fv i g) 8 : 2 v i 2 W 0 v i 62 W: We now have a standard linear programming relaxation of the traveling salesman problem, written for the k vertices in W . Results of Wolsey [16] and Shmoys and Williamson [14] show that if we apply Christofides s heuristic to produce a tour on the vertices in W , the length of this tour will be at most 3 2 times the optimum value z k t of this linear program. Thus, we obtain a tour on the vertices in W of length at most 3 2 Delta z k t 3(n Gamma m) 2(k ....

D. Shmoys and D. Williamson, "Analyzing the Held-Karp TSP bound: a monotonicity property with application," Information Proc. Letters, 35(1990), pp. 281--285.

....is not a welldefined number and is typically a moving target. Moreover, the HK bound by itself provides a reasonably close and consistent estimate of the optimal tour length. Worst case results tell us that, assuming the triangle inequality, the HK bound is at least 2 3 of the optimal tour length [31,37]. In practice, it appears to be much better, even in cases where the triangle inequality is violated. Let us consider the instance classes mentioned in the introduction, one by one. Random Euclidean Instances. This class, in which cities correspond to points uniformly distributed in the unit ....

D. B. SHMOYS AND D. P. WILLIAMSON, "Analyzing the HeldKarp TSP bound: A monotonicity property with applications, " Inform. Process. Lett. 35 (1990), 281-285.

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D. B. Shmoys and D. P. Williamson (1990). Analyzing the Held-Karp TSP bound: a monotonicity property with application. Information Proc. Lett. 35, 281--285.

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D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Info. Proc. Lett., 35(6):281-285, 1990.

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D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: A monotonicity property wtih application. IPL, 35:281--285, 1990.

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D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Information Proc. Lett., 35:281--285, 1991.

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