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arxiv: 2606.31530 · v1 · pith:IDEOS3BOnew · submitted 2026-06-30 · 💻 cs.DS · math.OC

Graph Scheduling with Group Completion Times

Lars Rohwedder , Leander Schnaars This is my paper

Pith reviewed 2026-07-01 02:32 UTC · model grok-4.3

classification 💻 cs.DS math.OC
keywords graph schedulinggroup completion timesiterated roundingapproximation algorithmcoflow schedulingdata migrationodd-set inequalitiesedge coloring
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The pith

Extending iterated rounding to general graphs yields a (2+ε)-approximation for edge scheduling with group completion times when OPT is large.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an iterated rounding technique previously developed for the bipartite case of coflow scheduling can be lifted to arbitrary multigraphs. The lift incorporates odd-set inequalities from the general matching polytope together with standard results on edge colorings of multigraphs. A sympathetic reader would care because the resulting guarantee is asymptotically tight and, when combined with an existing (1+φ) algorithm, produces a better ratio for the data-migration problem in every regime of OPT.

Core claim

By integrating odd-set inequalities and multigraph edge-coloring results into the iterated-rounding framework from the bipartite case, one obtains a (2+ε)-approximation algorithm for the general Graph Scheduling problem in the asymptotic regime where the optimal objective value is assumed to be large.

What carries the argument

Iterated rounding procedure that uses odd-set inequalities to control the matching polytope and invokes multigraph edge colorings to produce feasible integral schedules at each step.

If this is right

  • The new algorithm improves the best known approximation for Data Migration by returning the better of the (2+ε) guarantee and the prior (1+φ) guarantee in every regime.
  • The result is essentially tight for the asymptotic setting.
  • The same framework applies to both the coflow (bipartite) and data-migration (general) variants of the problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same integration of general-matching polytopes may be testable on other non-bipartite scheduling objectives that admit iterated-rounding analyses.
  • Efficient oracles for separating odd-set inequalities could be substituted to obtain faster practical implementations without changing the theoretical guarantee.

Load-bearing premise

The odd-set inequalities and edge-coloring results integrate into the iterated rounding loop without forcing extra loss factors or post-processing adjustments.

What would settle it

A family of multigraph instances with arbitrarily large OPT in which every schedule produced by the algorithm has cost strictly larger than (2+ε) times OPT.

read the original abstract

In the Graph Scheduling problem we schedule a given multiset of edges on discrete time steps, such that at each step the set of edges forms a matching. The goal is to minimize the sum of weighted group completion times, where a group is a set of edges and it completes when the last edge has been scheduled. Two popular variants of this problem are Coflow Scheduling and Data Migration. Our main result is extending a recent iterated rounding approach from Coflow Scheduling, roughly corresponding to the bipartite case, to the general Graph Scheduling problem. This yields an essentially tight $(2+\epsilon)$-approximation for the asymptotic setting where OPT is assumed to be large. For this we rely on polyhedral techniques from general matching, namely odd-set inequalities, and graph theoretical results on edge colorings in multigraphs. The state-of-the-art approximation algorithm for Data Migration is a $(1 + \phi)$-approximation that improves when OPT is small. Taking the best of this and our main result, we obtain an improvement of the approximation rate for Data Migration in any regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that by extending an iterated rounding approach previously used for Coflow Scheduling to the general Graph Scheduling problem, incorporating odd-set inequalities from the matching polytope and results on edge colorings of multigraphs, one obtains an essentially tight (2 + ε)-approximation algorithm in the asymptotic regime where the optimal value is large. Additionally, by combining this with the existing (1 + φ)-approximation for Data Migration, an improved approximation is achieved in all regimes.

Significance. If the central technical integration holds, the result would be significant for the field of scheduling and approximation algorithms, as it generalizes a recent technique to a broader class of problems while achieving a tight guarantee, and provides a practical improvement for the Data Migration problem.

major comments (2)
  1. The extension of the iterated rounding framework to incorporate odd-set inequalities is load-bearing for the (2+ε) guarantee. The analysis in the proof of the main theorem must explicitly demonstrate that adding these cutting planes does not change the rounding probabilities or the charging argument in a manner that introduces an approximation factor beyond (2+ε).
  2. It is unclear from the presentation how the multigraph edge-coloring results are used in conjunction with the polyhedral relaxation to maintain the asymptotic guarantee without additional loss.
minor comments (1)
  1. The abstract mentions 'essentially tight' but could specify the dependence on ε more clearly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The comments highlight areas where the presentation and proof details can be strengthened, and we address each point below with plans for revision.

read point-by-point responses

  1. Referee: The extension of the iterated rounding framework to incorporate odd-set inequalities is load-bearing for the (2+ε) guarantee. The analysis in the proof of the main theorem must explicitly demonstrate that adding these cutting planes does not change the rounding probabilities or the charging argument in a manner that introduces an approximation factor beyond (2+ε).

    Authors: We agree that explicit verification is required. The iterated rounding is performed on a basic solution to the LP that lies in the matching polytope (including odd-set inequalities). The rounding probabilities derive from the fact that every basic solution has a variable of value at least 1/2 (or the support allows selection without violating the polytope), and the charging argument bounds the expected group completion by twice the fractional value plus lower-order terms; the odd-set constraints are satisfied by construction and do not alter these probabilities or the factor-2 bound. To make this fully transparent we will insert a short lemma immediately preceding the main theorem that isolates the effect of the odd-set inequalities and confirms they introduce no extra multiplicative loss beyond (2+ε). revision: yes

  2. Referee: It is unclear from the presentation how the multigraph edge-coloring results are used in conjunction with the polyhedral relaxation to maintain the asymptotic guarantee without additional loss.

    Authors: The edge-coloring results (specifically, the bound on the chromatic index of multigraphs) are invoked after the iterated-rounding step produces an integral matching whose total fractional weight is at most (2+ε) times the LP value. Because the LP value is at most OPT and the coloring schedules each matching in at most Δ+1 colors, the actual makespan is (1+o(1)) times the rounded value when OPT is large; this yields the asymptotic (2+ε) guarantee with no further loss. We will revise the exposition in Section 4 to include an explicit pipeline diagram and a paragraph that traces the polyhedral solution through rounding and coloring, making the absence of additional loss clear. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central extension rests on external polyhedral and coloring results

full rationale

The derivation chain begins from the standard matching polytope (odd-set inequalities) and Vizing's theorem for multigraph edge colorings, both external and independently established. The iterated rounding framework is extended from the bipartite coflow case to general graphs, with the (2+ε) guarantee obtained via a new charging argument that incorporates these tools without reducing the target bound to any fitted parameter or self-referential definition. No equation or claim in the abstract or described approach equates the output approximation ratio to an input by construction, and the self-contained algorithmic analysis does not rely on a load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from matching theory and graph theory. No free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Odd-set inequalities define the matching polytope for general graphs
    Invoked to extend the rounding approach beyond bipartite graphs.
  • standard math Multigraphs admit edge colorings with a bounded number of colors
    Used to schedule the rounded edges into matchings.

pith-pipeline@v0.9.1-grok · 5709 in / 1395 out tokens · 47407 ms · 2026-07-01T02:32:41.002976+00:00 · methodology

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