Paired approximation problems and incompatible inapproximabilities

TL;DR

This paper explores paired approximation strategies for related optimization problems, achieving better combined approximations in some cases, but also identifying limitations where no such improvements are possible.

Contribution

It introduces new paired approximation algorithms for certain problem pairs and demonstrates inherent limitations for others, advancing understanding of inapproximability tradeoffs.

Findings

Efficient paired approximation for (1,2)-TSP and maximum independent set.

Paired approximation for coloring or long path problems.

Limitations for set cover/hitting set and clique/independent set pairs.

Abstract

This paper considers pairs of optimization problems that are defined from a single input and for which it is desired to find a good approximation to either one of the problems. In many instances, it is possible to efficiently find an approximation of this type that is better than known inapproximability lower bounds for either of the two individual optimization problems forming the pair. In particular, we find either a $(1+\epsilon)$-approximation to $(1,2)$-TSP or a $1/\epsilon$-approximation to maximum independent set, from a given graph, in linear time. We show a similar paired approximation result for finding either a coloring or a long path. However, no such tradeoff exists in some other cases: for set cover and hitting set problems defined from a single set family, and for clique and independent set problems on the same graph, it is not possible to find an approximation when both problems are combined that is better than the best approximation for either problem on its own.