Weighted amplifiers and inapproximability results for Travelling Salesman problem

The 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.

The sharp threshold for percolation on expander graphs

We consider a random subgraph G n(p) of a finite graph family G n = (V n, E n) formed by retaining each edge of G n independently with probability p. We show that if G n is an expander graph with vertices of bounded degree, then for any c n ∈ (0, 1) satisfying $$c_n \gg {1 \mathord{\left/ {\vphantom {1 {\sqrt {\ln n} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\ln n} }}$$ and $$\mathop {\lim \sup }\limits_{n \to \infty } c_n < 1$$, the property that the random subgraph contains a giant component of order c n|V n| has a sharp threshold.