Polynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions

TL;DR

This paper extends polynomial-time data reduction techniques to a broad class of weighted problems with non-additive goal functions, enabling efficient kernelization for various complex optimization problems.

Contribution

It characterizes the function types suitable for Frank and Tardos' technique, broadening its applicability beyond additive functions, and applies this to multiple problem domains.

Findings

Polynomial-size kernels for a wide range of weighted problems.

Generalization of kernelization results to non-additive goal functions.

Improved and unified kernelization techniques across multiple problem areas.

Abstract

Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of a parameter chosen in advance. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We characterize the function types that the technique is applicable to, which turns out to contain many non-additive functions. Using this insight, we systematically obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.