A Randomized Rounding Approach to the Traveling Salesman Problem

Abstract

For some positive constant 0 , we give a ( 3 2 − 0 )-approximation algorithm for the following problem: given a graph G 0 = (V, E 0 ), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G 0 . The result improves on the 3 2 -approximation algorithm due to Christofides [13] for this special case.Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation.Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.

TSP on Cubic and Subcubic Graphs

“…Surprisingly, in over three decades no one has found an approximation algorithm which improves upon this bound of 3/2, even for the special case of graph-TSP, and the quest for finding such improvements is one of the most challenging research questions in combinatorial optimization. Very recently, Gharan et al [16] announced a randomized 3/2 − approximation for graph-TSP for some > 0.…”

Near-Linear Time Approximation Schemes for some Implicit Fractional Packing Problems

“…Algorithms for decomposing a point in the spanning tree polytope into a convex combination of spanning trees have been used in recent algorithms for approximating TSP in both undirected and directed graphs [4,17] (and several subsequent papers). Connections to computing minimum cuts: The first step in the near-linear-time randomized global minimum cut algorithm of Karger [21] is to find a 1− 2 -approximate spanning tree packing with O(log n) trees in the packing.…”

Aplikasi Hybrid Firefly Algorithm untuk Pemecahan Masalah Traveling Salesman: Studi Kasus pada PT Anugerah Mandiri Success

“…TSP sendiri merupakan topik yang sangat menarik di bidang optimalisasi dan masih hangat diperbincangkan (Tadei, Perboli, & Perfetti, 2013). Akan tetapi, permasalahan optimalisasi yang akan diangkat tidaklah mudah untuk dapat diselesaikan (Gharan, Saberi, & Singh, 2011). Pokok permasalahan di sini ialah bagaimana meminimalkan total biaya di mana total biaya merupakan hasil dari perhitungan biaya yang dibutuhkan untuk melakukan perjalanan dari suatu tempat dengan melewati semua tempat tujuan tepat satu kali dan kembali ke tempat semula (Rego, Gamboa, Glover, & Osterman, 2011).…”