Abstract
AbstractThe expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from $$\frac{123}{122}$$
123
122
(Karpinski et al. in J Comput Syst Sci 81(8):1665–1677, 2015) to $$\frac{117}{116}$$
117
116
. This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Theory and Mathematics,Control and Optimization,Discrete Mathematics and Combinatorics,Computer Science Applications
Reference17 articles.
1. Böckenhauer HJ, Hromkovič J, Klasing R, Seibert S, Unger W (2000) An improved lower bound on the approximability of metric TSP and approximation algorithms for the TSP with sharpened triangle inequality. In: Proceedings of the 17th annual symposium on theoretical aspects of computer science , STACS’00. Springer, Berlin, pp 382—394
2. Chlebík M, Chlebíková J (2003) Approximation hardness for small occurrence instances of NP-hard problems. In: Algorithms and complexity, 5th Italian conference, CIAC 2003, Lecture Notes in Computer Science, vol 2653, pp 152–164
3. Chlebík M, Chlebíková J (2008) The Steiner Tree problem on graphs: inapproximability results. Theor Comput Sci 406(3):207–214
4. Chlebík M, Chlebíková J (2019) Approximation hardness of Travelling Salesman via weighted amplifiers. In: Computing and combinatorics—25th international conference, COCOON 2019, Springer, Lecture Notes in Computer Science, vol 11653, pp 115–127
5. Christofides N (1976) Worst-case analysis of a new heuristic for the Travelling Salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University