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 \(\frac{123}{122}\) (Karpinski et al. in J Comput Syst Sci 81(8):1665–1677, 2015) to \(\frac{117}{116}\). 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.

扩展器图结构及其变体是减少保留间隙以证明组合优化问题的近似下界的主要工具。在本文中，我们介绍了加权放大器和约束满足问题的加权低发生率，作为NP硬间隙减少的中间步骤。对于边缘加权问题（例如Traveling Salesman或Steiner Tree），在中间问题中允许权重是很自然的。我们向旅行推销员演示该技术并在间隙减小中使用参数化加权放大器，以便在微调其扩展参数时具有更大的灵活性。本文的目的是指出这些想法的有效性，而不是优化扩展器的参数。尽管如此，我们表明，已知膨胀剂值的已经稍有改善，就会适度地从\（\ frac {123} {122} \）改善TSP当前的最佳近似硬度值（Karpinski等人，J Comput Syst Sci 81（8）： 1665–1677，2015）到\（\ frac {117} {116} \）。这为研究随机图的扩展性质以改善TSP的近似下限和其他边缘加权优化问题提供了新的动力。