Tight Approximation Bounds for the Seminar Assignment Problem

TL;DR

This paper introduces a greedy algorithm for the seminar assignment problem, providing a tight approximation bound of (1 - e^{-1}) and establishing its optimality under standard complexity assumptions.

Contribution

The paper proves that a natural greedy algorithm achieves a (1 - e^{-1}) approximation ratio for the seminar assignment problem, matching the best possible bound unless unlikely complexity class collapses.

Findings

Greedy algorithm achieves (1 - e^{-1}) approximation ratio.

This bound is proven to be tight under standard complexity assumptions.

The problem is NP-complete and lacks a PTAS.

Abstract

The seminar assignment problem is a variant of the generalized assignment problem in which items have unit size and the amount of space allowed in each bin is restricted to an arbitrary set of values. The problem has been shown to be NP-complete and to not admit a PTAS. However, the only constant factor approximation algorithm known to date is randomized and it is not guaranteed to always produce a feasible solution. In this paper we show that a natural greedy algorithm outputs a solution with value within a factor of $(1 - e^{-1})$ of the optimal, and that unless $NP\subseteq DTIME(n^{\log\log n})$, this is the best approximation guarantee achievable by any polynomial time algorithm.