The Unbounded Integrality Gap of a Semidefinite Relaxation of the Traveling Salesman Problem
Abstract
We study a semidefinite programming relaxation of the traveling salesman problem introduced by de Klerk, Pasechnik, and Sotirov [SIAM J. Optim., 19 (2008), pp. 1559--1573] and show that their relaxation has an unbounded integrality gap. In particular, we give a family of instances such that the gap increases linearly with $n$. To obtain this result, we search for feasible solutions within a highly structured class of matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, these solutions imply several corollaries that help us better understand the semidefinite program and its relationship to other TSP relaxations. Using the same technique, we show that a more general semidefinite program introduced by de Klerk, de Oliveira Filho, and Pasechnik [Handbook on Semidefinite, Conic and Polynomial Optimization, Springer, New York, 2012, pp. 171--199.] for the $k$-cycle cover problem also has an unbounded integrality gap.
References
[1]
S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, J. ACM, 45 (1998), pp. 753--782.
[2]
K. K. Cheung, On Lovász--Schrijver lift-and-project procedures on the Dantzig--Fulkerson--Johnson relaxation of the TSP, SIAM J. Optim., 16 (2005), pp. 380--399.
[3]
E. Chlamtac and M. Tulsiani, Convex relaxations and integrality gaps, in Handbook on Semidefinite, Conic and Polynomial Optimization, Internat. Ser. Oper. Res. Management Sci. 166, M. F. Anjos and J. B. Lasserre, eds., Springer, New York, 2012, pp. 139--169.
[4]
N. Christofides, Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem, Technical report 388, Carnegie-Mellon University, Pittsburgh, PA, 1976.
[5]
D. Cvetković, M. Čangalović, and V. Kovačević-Vujčić, Semidefinite programming methods for the symmetric traveling salesman problem, in IPCO 1999: Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 1610, G. Cornuéjols, R. E. Burkard, and G. J. Woeginger, eds., Springer, Berlin, 1999, pp. 126--136.
[6]
G. Dantzig, R. Fulkerson, and S. Johnson, Solution of a large-scale traveling-salesman problem, J. Oper. Res. Soc. Amer., 2 (1954), pp. 393--410.
[7]
E. de Klerk, F. de Oliveira Filho, and D. Pasechnik, Relaxations of combinatorial problems via association schemes, in Handbook on Semidefinite, Conic and Polynomial Optimization, Internat. Ser. Oper. Res. Management Sci. 166, M. F. Anjos and J. B. Lasserre, eds., Springer, New York, 2012, pp. 171--199.
[8]
E. de Klerk and C. Dobre, A comparison of lower bounds for the symmetric circulant traveling salesman problem, Discrete Appl. Math., 159 (2011), pp. 1815--1826.
[9]
E. de Klerk, D. V. Pasechnik, and R. Sotirov, On semidefinite programming relaxations of the traveling salesman problem, SIAM J. Optim., 19 (2008), pp. 1559--1573.
[10]
E. de Klerk and R. Sotirov, Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry, Math. Program., 133 (2012), pp. 75--91.
[11]
Z. Gao, On the metric $s$--$t$ path traveling salesman problem, SIAM J. Discrete Math., 29 (2015), pp. 1133--1149.
[12]
S. O. Gharan, A. Saberi, and M. Singh, A randomized rounding approach to the traveling salesman problem, in Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2011, pp. 550--559; available at https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6108120.
[13]
M. Goemans and F. Rendl, Combinatorial optimization, in Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Internat. Ser. Oper. Res. Management Sci. 27, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Springer, New York, 2000, pp. 343--360.
[14]
M. X. Goemans and F. Rendl, Semidefinite programs and association schemes, Computing, 63 (1999), pp. 331--340.
[15]
M. X. Goemans and D. P. Williamson, A general approximation technique for constrained forest problems, SIAM J. Computing, 24 (1995), pp. 296--317.
[16]
R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inform. Theory, 2 (2006), pp. 155--239.
[17]
M. Grötschel and M. W. Padberg, Polyhedral theory, in The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan, and D. B. Shmoys, eds., John Wiley & Sons, New York, 1985, pp. 251--306.
[18]
S. C. Gutekunst and D. P. Williamson, The Unbounded Integrality Gap of a Semidefinite Relaxation of the Traveling Salesman Problem, preprint, https://arxiv.org/abs/1710.08455, 2017.
[19]
M. Held and R. M. Karp, The traveling-salesman problem and minimum spanning trees, Oper. Res., 18 (1970), pp. 1138--1162.
[20]
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
[21]
A. Jeffrey and H.-H. Dai, Handbook of Mathematical Formulas and Integrals, 4th ed., Academic Press, New York, 2008.
[22]
M. Karpinski, M. Lampis, and R. Schmied, New inapproximability bounds for TSP, J. Comput. System Sci., 81 (2015), pp. 1665--1677.
[23]
J. B. Lasserre, An explicit exact SDP relaxation for nonlinear 0--1 programs, in IPCO 2001: Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 2081, K. Aardal and B. Gerards, eds., Springer, Berlin, 2001, pp. 293--303.
[24]
L. Lovász and A. Schrijver, Cones of matrices and set-functions and 0--1 optimization, SIAM J. Optim., 1 (1991), pp. 166--190.
[25]
J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, $k$-MST, and related problems, SIAM J. Comput., 28 (1999), pp. 1298--1309.
[26]
T. Mömke and O. Svensson, Removing and adding edges for the traveling salesman problem, J. ACM, 63 (2016), Article 2.
[27]
M. Mucha, 13/9-approximation for graphic TSP, Theory Comput. Syst., 55 (2014), pp. 640--657.
[28]
A. Sebö and J. Vygen, Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs, Combinatorica, 34 (2014), pp. 597--629.
[29]
H. D. Sherali and W. P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM J. Discrete Math., 3 (1990), pp. 411--430.
[30]
D. B. Shmoys and D. P. Williamson, Analyzing the Held--Karp TSP bound: A monotonicity property with application, Inform. Process. Lett., 35 (1990), pp. 281--285.
[31]
D. A. Spielman, Spectral graph theory and its applications, in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2007, pp. 29--38; available at https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4389466.
[32]
D. P. Williamson and D. B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, New York, 2011.
[33]
L. A. Wolsey, Heuristic analysis, linear programming and branch and bound, in Combinatorial Optimization II, V. J. Rayward-Smith, ed., Springer, Berlin, 1980, pp. 121--134.
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DOI:10.1137/sjope8.28.3
Issue’s Table of Contents© 2018, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2018
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