Weighted Chairman Assignment and Flow-Time Scheduling

TL;DR

This paper extends the chairman assignment problem to weighted cases, proves a related flow conjecture, and applies these results to develop a 3-approximation algorithm for a complex scheduling problem involving release and closing times.

Contribution

It generalizes the chairman assignment problem to weighted scenarios, confirms a special case of a flow conjecture, and introduces a new approximation algorithm for a scheduling problem.

Findings

Existence of a weighted assignment close to a fractional solution.

Confirmation of a special case of the flow conjecture.

Development of a 3-approximation algorithm for flow-time minimization.

Abstract

Given positive integers $m, n$, a fractional assignment $x \in [0,1]^{m \times n}$ and weights $d \in \mathbb{R}^n_{>0}$, we show that there exists an assignment $y \in \{0,1\}^{m \times n}$ so that for every $i\in[m]$ and $t\in [n]$, \[ \Big|\sum_{j \in [t]} d_j (x_{ij} - y_{ij}) \Big| < \max_{j \in [n]} d_j. \] This generalizes a result of Tijdeman (1973) on the unweighted version, known as the chairman assignment problem. This also confirms a special case of the single-source unsplittable flow conjecture with arc-wise lower and upper bounds due to Morell and Skutella (IPCO 2020). As an application, we consider a scheduling problem where jobs have release times and machines have closing times, and a job can only be scheduled on a machine if it is released before the machine closes. We give a $3$-approximation algorithm for maximum flow-time minimization.