Polytope Scheduling with Groups: Unified Models and Optimal Guarantees

We propose new abstract and unified perspectives on a range of scheduling and graph coloring problems with general min-sum objectives. Specifically, we consider various problems where the objective function is the weighted sum of completion times over groups of entities (jobs, vertices, or edges), thereby generalizing two important objectives in scheduling: makespan and the sum of weighted completion times. As one of our main results, we present a best-possible $\mathcal O(\log g)$-competitive algorithm in the non-clairvoyant online setting, where $g$ denotes the size of the largest group. This is the first non-trivial competitive bound for several problems with group completion time objective, and it is an exponential improvement over previous results for non-clairvoyant coflow scheduling. For offline scheduling, we provide elegant yet powerful meta-frameworks that, in a unifying way, yield new or stronger approximation algorithms for our new abstract problems as well as for previously well-studied special cases.

💡 Research Summary

The paper introduces a unifying abstraction for a wide range of scheduling and graph‑coloring problems in which the objective is the weighted sum of group completion times. A “group” is any subset of entities (jobs, vertices, or edges) and each group S carries a weight w_S > 0. The completion time of a group is defined as C_S = max_{j∈S} C_j, where C_j is the completion time of entity j. The goal is to minimize Σ_{S∈𝒮} w_S C_S. This formulation simultaneously generalizes makespan (all jobs in a single group) and the classic Σw_j C_j objective (each job forms its own group).

Two abstract models are defined. In Polytope Scheduling with Groups (PSP‑G), time is continuous and at each instant a processing‑rate vector y(t) must be chosen from a down‑closed polytope P = {y ≥ 0 | B y ≤ 1}. The entry y_j(t) denotes the instantaneous processing allocated to job j; a job finishes when the integral of y_j(t) reaches its processing requirement p_j. The objective remains Σ w_S max_{j∈S} C_j. The Discrete Polytope Scheduling with Groups (DPSP‑G) is the discrete analogue: a finite set of feasible points P′⊆P is given, each job receives a constant rate during a contiguous interval