Graph-TSP from Steiner Cycles

We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph \(G\), if we can find a spanning tree \(T\) and a simple cycle that contains the vertices with odd-degree in \(T\), then we show how to combine the classic “double spanning tree” algorithm with Christofides’ algorithm to obtain a TSP tour of length at most \(\frac{4n}{3}\). We use this approach to show that a graph containing a Hamiltonian path has a TSP tour of length at most \(4n/3\).

[1]  Uppaluri S. R. Murty,et al.  Graph Theory with Applications , 1978 .

[2]  Juan José Salazar González,et al.  The Steiner cycle polytope , 2003, European Journal of Operational Research.

[3]  G. Dirac.  Some Theorems on Abstract Graphs , 1952 .

[4]  Juan-José Salazar-González.  The Steiner cycle polytope , 2003 .

[5]  Tudor Zamfirescu,et al.  Three small cubic graphs with interesting hamiltonian properties , 1980, J. Graph Theory.

[6]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[7]  Marcin Mucha.  13/9-approximation for Graphic TSP , 2012, STACS.

[8]  Brendan D. McKay,et al.  A nine point theorem for 3-connected graphs , 1982, Comb..

[9]  Kenta Ozeki,et al.  A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph , 2008, Graphs Comb..

[10]  M. Steinová.  Approximability of the Minimum Steiner Cycle Problem , 2012 .

[11]  Ola Svensson,et al.  Approximating Graphic TSP by Matchings , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[12]  Nisheeth K. Vishnoi.  A Permanent Approach to the Traveling Salesman Problem , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[13]  Leen Stougie,et al.  TSP on Cubic and Subcubic Graphs , 2011, IPCO.

[14]  Paul Erdös,et al.  A note on Hamiltonian circuits , 1972, Discret. Math..

[15]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[16]  G. Dirac.  SHORT PROOF OF MENGER'S GRAPH THEOREM , 1966 .

[17]  A. J.,et al.  Analysis of Christofides ' heuristic : Some paths are more difficult than cycles , 2002 .

[18]  Gérard Cornuéjols,et al.  The traveling salesman problem on a graph and some related integer polyhedra , 1985, Math. Program..

[19]  Brendan D. McKay,et al.  Cycles Through 23 Vertices in 3-Connected Cubic Planar Graphs , 1999, Graphs Comb..

[20]  Samir Khuller,et al.  Approximation Algorithms with Bounded Performance Guarantees for the Clustered Traveling Salesman Problem , 1998, FSTTCS.

[21]  András Sebő.  Eight-Fifth approximation for the path TSP , 2013, IPCO 2013.

[22]  Jens Vygen,et al.  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 , 2012, ArXiv.

[23]  András Sebö,et al.  Eight-Fifth Approximation for the Path TSP , 2012, IPCO.

[24]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[26]  Mohit Singh,et al.  A Randomized Rounding Approach to the Traveling Salesman Problem , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[27]  Robert D. Carr,et al.  Towards a 4/3 approximation for the asymmetric traveling salesman problem , 1999, SODA '00.

[28]  Shi Ronghua,et al.  2-neighborhoods and Hamiltonian conditions , 1992 .

[29]  阿部 浩一,et al.  Fundamenta Mathematicae私抄 : 退任の辞に代えて , 1987 .

[30]  Marcin Mucha.  $\frac{13}{9}$-Approximation for Graphic TSP , 2014, Theory Comput. Syst..