A domination algorithm for {0,1}\{0,1\}-instances of the travelling salesman problem

We present an approximation algorithm for $\{0,1\}$-instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio $1-n^{-1/29}$. In other words, given a $\{0,1\}$-edge-weighting of the complete graph $K_n$ on $n$ vertices, our algorithm outputs a Hamilton cycle $H^*$ of $K_n$ with the following property: the proportion of Hamilton cycles of $K_n$ whose weight is smaller than that of $H^*$ is at most $n^{-1/29}$. Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio $1/2-o(1)$ for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant $C$ such that $n^{-1/29}$ cannot be replaced by $\exp(-(\log n)^C)$ in the result above.Comment: 29 pages (final version to appear in Random Structures and Algorithms

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

How to make a greedy heuristic for the asymmetric traveling salesman problem competitive

It is widely confirmed by many computational experiments that a greedy type heuristics for the Traveling Salesman Problem (TSP) produces rather poor solutions except for the Euclidean TSP. The selection of arcs to be included by a greedy heuristic is usually done on the base of cost values. We propose to use upper tolerances of an optimal solution to one of the relaxed Asymmetric TSP (ATSP) to guide the selection of an arc to be included in the final greedy solution. Even though it needs time to calculate tolerances, our computational experiments for the wide range of ATSP instances show that tolerance based greedy heuristics is much more accurate an faster than previously reported greedy type algorithms