Duality between quasi-concave functions and monotone linkage functions

TL;DR

This paper explores the relationship between quasi-concave functions and monotone linkage functions, showing how certain quasi-concave functions can be optimized efficiently and analyzing their properties across different set families.

Contribution

It provides a detailed analysis of quasi-concave functions on various set families and investigates their duality with monotone linkage functions, extending previous work on antimatroids.

Findings

Quasi-concave functions can be optimized in polynomial time when defined via monotone linkage functions.

A correspondence between quasi-concave functions and bottleneck functions is established on antimatroids.

The paper extends the analysis of quasi-concave functions beyond antimatroids to other set families.

Abstract

A function $F$ defined on all subsets of a finite ground set $E$ is quasi-concave if $F(X\cup Y)\geq\min\{F(X),F(Y)\}$ for all $X,Y\subset E$. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concave function is defined by associated monotone linkage function then it can be optimized by the greedy type algorithm in a polynomial time. Quasi-concave functions defined as minimum values of monotone linkage functions were considered on antimatroids, where the correspondence between quasi-concave and bottleneck functions was shown (Kempner & Levit, 2003). The goal of this paper is to analyze quasi-concave functions on different families of sets and to investigate their relationships with monotone linkage functions.