Fair Repetitive Interval Scheduling

TL;DR

The paper addresses fair resource allocation in repetitive, single‑machine scheduling across $m$ days for $n$ clients, aiming to guarantee each client at least $k$ scheduled days under interval constraints $ (d_{i,j}-p_{i,j},d_{i,j}]$. It establishes a dichotomy: the problem is polynomial when $k\in\{0,m-1,m\}$ and NP-hard otherwise, while also exploring unit processing times and day‑independent data. It develops a spectrum of algorithmic tools, including 2‑SAT reductions, bipartite matching, DP on nice tree decompositions, and ILP with Lenstra’s theorem, to obtain FPT results parameterized by $(m+\tau)$ and by the number of clients $n$, respectively. The work leverages conflict graphs (daily interval graphs) and their treewidth to study structural parameters, and it discusses generalizations (e.g., client‑dependent fairness, multiple machines) and open questions that guide future research in fair repetitive scheduling. These results illuminate where efficient fair schedules are feasible and provide concrete algorithmic techniques for practical JIT-style scheduling under fairness constraints.

Abstract

Fair resource allocation is undoubtedly a crucial factor in customer satisfaction in several scheduling scenarios. This is especially apparent in repetitive scheduling models where the same set of clients repeatedly submits jobs on a daily basis. In this paper, we aim to analyze a repetitive scheduling system involving a set of $n$ clients and a set of $m$ days. On every day, each client submits a request to process a job exactly within a specific time interval, which may vary from day to day, modeling the scenario where the scheduling is done Just-In-Time (JIT). The daily schedule is executed on a single machine that can process a single job at a time, therefore it is not possible to schedule jobs with intersecting time intervals. Accordingly, a feasible solution corresponds to sets of jobs with disjoint time intervals, one set per day. We define the quality of service (QoS) that a client receives as the number of executed jobs over the $m$ days period. Our objective is to provide a feasible solution where each client has at least $k$ days where his jobs are processed. We prove that this problem is NP-hard even under various natural restrictions such as identical processing times and day-independent due dates. We also provide efficient algorithms for several special cases and analyze the parameterized tractability of the problem with respect to several parameters, providing both parameterized hardness and tractability results.

Figures (8)

• Figure 1: Day 1.
• Figure 2: Day 2.
• Figure 3: Day 3.
• Figure 4: The construction of a bipartite graph from an instance of $1\mid\mathop{\mathrm{rep}}\nolimits, p_{i,j}=1\mid\min_j \sum_i Z_{i,j}$ with $m=3$ and $k=1$, where only the relevant vertices for two clients $j$ and $j'$ are depicted. Clients $j$ and $j'$ have different due date on day 1, and the same due date on day 2 and 3. Both jobs of client $j$ and $j'$ have two rejection vertices with which they can also be potentially matched.
• Figure 5: An example for schedules on the vertex selection days and the validation days given a multicolored independent set $I$. We mark in bold the jobs which are scheduled; the jobs not appearing in the gadget are all identical to the job of client $c_0$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5
• Definition 6
• Theorem 7
• proof
• Corollary 8
• Theorem 9