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Weighted Chairman Assignment and Flow-Time Scheduling

Siyue Liu, Victor Reis

TL;DR

This work introduces the weighted chairman assignment problem, proving that for any fractional $m\times n$ assignment $x$ and positive weights $d$, one can efficiently compute an integral $y$ with prefix discrepancy bounded by $\left(1-\frac{1}{2m-2}\right)\max_j d_j$. This generalizes the classical chairman problem and yields a concrete link to the Morell-Skutella unsplittable flow conjecture, providing a linear-time algorithm and a 3-approximation for maximum flow-time minimization on machines with closing times. The paper also develops tight lower bounds, shows limits of the approach for interval bounds, and outlines open questions on non-uniform weights and broader discrepancy settings. Together these results advance prefix-discrepancy techniques and their applications to scheduling and unsplittable flow problems with nonuniform demands.

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.

Weighted Chairman Assignment and Flow-Time Scheduling

TL;DR

This work introduces the weighted chairman assignment problem, proving that for any fractional assignment and positive weights , one can efficiently compute an integral with prefix discrepancy bounded by . This generalizes the classical chairman problem and yields a concrete link to the Morell-Skutella unsplittable flow conjecture, providing a linear-time algorithm and a 3-approximation for maximum flow-time minimization on machines with closing times. The paper also develops tight lower bounds, shows limits of the approach for interval bounds, and outlines open questions on non-uniform weights and broader discrepancy settings. Together these results advance prefix-discrepancy techniques and their applications to scheduling and unsplittable flow problems with nonuniform demands.

Abstract

Given positive integers , a fractional assignment and weights , we show that there exists an assignment so that for every and , \[ \Big|\sum_{j \in [t]} d_j (x_{ij} - y_{ij}) \Big| < \max_{j \in [n]} d_j. \] 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 -approximation algorithm for maximum flow-time minimization.

Paper Structure

This paper contains 9 sections, 12 theorems, 32 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization
  3. Related work
  4. Proof of Theorem \ref{['thm:WCAP']}
  5. Application for flow-time scheduling
  6. Lower bounds
  7. Open problems
  8. Acknowledgements
  9. Computer program

Key Result

Theorem 1

Given positive integers $m, n$ with $m > 1$, a fractional assignment $x \in [0,1]^{m \times n}$ and weights $d \in \mathbb{R}^n_{>0}$, there is a linear-time algorithm that computes an assignment $y \in \{0,1\}^{m \times n}$ so that for every $i\in[m]$ and $t\in [n]$,

Figures (2)

  • Figure 1: Reduction for $m = 4$, $n = 3$. We create $n$ copies $i_1, \dots, i_n$ of each $i \in [m]$ to capture all prefixes. Terminal $t_j$ has demand $\sum_{i\in[m]} x_{ij}d_j=d_j$, and is connected to $i_j$ for each $i\in [m]$. The flow value on arc $(i_{t+1},i_t)$ is $\sum_{j\in[t]} x_{ij}d_j$.
  • Figure 2: There are $m$ machines $M=[m]$ with closing times $b_i=i\cdot \delta$ for some $\delta \ll \tfrac{1}{m}$. There are $m$ batches of jobs, released at times $\delta, 2\delta,...,m\delta$. The $j$-th batch has $(m-j+1)$ jobs, each with processing time $\frac{1}{m-j+1}$. Left: FIFO schedules the $j$-th batch to machines $j,j+1,...,m$, one for each, because those are the machines that are not closed yet. The maximum flow-time is $(\sum_{j\in [m]} \tfrac{1}{j}) - m\delta=\Omega(\log m)$. Right: OPT schedules the $j$-th batch to machine $j$, which is feasible because the $j$-th batch is released no later than machine $j$ is closed. The maximum flow-time is $1-m\delta \le 1$.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Proposition 4
  • Proposition 5
  • Conjecture 1: MorellSkutella2020IPCO
  • Conjecture 2
  • Lemma 6
  • proof
  • ...and 28 more