Conference Program Design with Single-Peaked and Single-Crossing Preferences

Abstract

We consider the Conference Program Design (CPD) problem, a multi-round generalization of (the maximization versions of) q-Facility Location and the Chamberlin-Courant multi-winner election, introduced by (Caragiannis, Gourvès and Monnot, IJCAI 2016). CPD asks for the selection of kq items and their assignment to k disjoint sets of size q each. The agents receive utility only from their best item in each set and we want to maximize the total utility derived by all agents from all sets. Given that CPD is $\mathbf {NP}$-hard for general utilities, we focus on utility functions that are either single-peaked or single-crossing. For general single-peaked utilities, we show that CPD is solvable in polynomial time and that Percentile Mechanisms are truthful. If the agent utilities are given by distances in the unit interval, we show that a Percentile Mechanism achieves an approximation ratio 1 / 3, if $q=1$, and at least $(2q-3)/(2q-1)$, for any $q \ge 2$. On the negative side, we show that a generalization of CPD, where some items must be assigned to specific sets in the solution, is $\mathbf {NP}$-hard for dichotomous single-peaked preferences. For single-crossing preferences, we present a dynamic programming exact algorithm that runs in polynomial time if k is constant.

Supported by the projects ANR CoCoRICo-CoDec and PSL MULTIFAC.