Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions

TL;DR

This paper introduces and analyzes a new subclass of submodular functions called discounted price functions, providing tight bounds and novel greedy algorithms for fundamental combinatorial optimization problems.

Contribution

It defines discounted price functions, extends optimization algorithms to this class, and develops adaptive greedy algorithms with proven bounds for key problems.

Findings

Tight upper and lower bounds for Edge Cover, Spanning Tree, Perfect Matching, Shortest Path.

Novel adaptive greedy algorithms that rectify previous mistakes.

Generalization of linear cost functions with efficient representations.

Abstract

Submodular functions are an important class of functions in combinatorial optimization which satisfy the natural properties of decreasing marginal costs. The study of these functions has led to strong structural properties with applications in many areas. Recently, there has been significant interest in extending the theory of algorithms for optimizing combinatorial problems (such as network design problem of spanning tree) over submodular functions. Unfortunately, the lower bounds under the general class of submodular functions are known to be very high for many of the classical problems. In this paper, we introduce and study an important subclass of submodular functions, which we call discounted price functions. These functions are succinctly representable and generalize linear cost functions. In this paper we study the following fundamental combinatorial optimization problems: Edge Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight upper and lower bounds for these problems. The main technical contribution of this paper is designing novel adaptive greedy algorithms for the above problems. These algorithms greedily build the solution whist rectifying mistakes made in the previous steps.