Analyzing the Held-Karp TSP Bound: A Monotonicity Property with Application

In their 1971 paper on the Traveling Salesman Problem and Minimum Spanning Trees, Held and Karp showed that finding an optimally weighted 1-tree is equivalent to solving a linear program for the Traveling Salesman Problem (TSP) with only node-degree constraints and subtour elimination constraints. In this paper we show that the Held-Karp 1-trees have a certain monotonicity property: given a particular instance of the symmetric TSP with triangle inequality, the cost of the minimum weighted 1-tree is monotonic with respect to the set of nodes included. As a consequence, we obtain an alternate proof of a result of Wolsey and show that linear programs with node-degree and subtour elimination constraints must have a cost at least 23OPT , where OPT is the cost of the optimum solution to the TSP instance. The traveling salesman problem is one of the most notorious in the field of combinatorial optimization, and one of the most well-studied [7]. Currently, the most successful approach to finding optimal solutions to large-scale problems is based on formulating the problem as a linear program and finding explicit partial descriptions of this linear polytope [5], [8]. The most natural constraints are derived from an integer linear programming formulation that uses nodedegree constraints and subtour elimination constraints. We focus our attention on symmetric instances of the TSP that obey the triangle inequality. Let V = {1, 2, . . . , n} denote the set of nodes. For any distinct i and j, assign a cost cij such that cij = cji, and for any k distinct from i and j, cij ≤ cik + ckj . Then the Subtour LP on this instance is B = min ∑ 1≤ii xij + ∑ j<i xji = 2, i = 1, 2, . . . , n, ∑ i∈S,j∈S,i<j xij ≤ |S|− 1, for any proper subsetS ⊂ V, 0 ≤ xij ≤ 1, 1 ≤ i < j ≤ n. (1) ∗This research was supported in part by Air Force Contract AFOSR-86-0078 and by the National Science Foundation under a Presidential Young Investigator Award CCR-8996272 with matching support from IBM, Sun, and UPS.