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arxiv: 2604.13642 · v1 · submitted 2026-04-15 · 💻 cs.DS

Lawler-Moore Speedups via Additive Combinatorics

Karl Bringmann , Danny Hermelin , Tomohiro Koana , Dvir Shabtay This is my paper

Pith reviewed 2026-05-10 12:16 UTC · model grok-4.3

classification 💻 cs.DS
keywords parallel machine schedulingdynamic programmingLawler-Mooreadditive combinatoricsstate pruningscheduling objectivesload balancing
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The pith

A swapping argument bounds load differences to 4p_max squared in optimal schedules, replacing P with p_max squared in Lawler-Moore dynamic programming.

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

The paper improves the classical Lawler-Moore dynamic program for scheduling jobs on m identical parallel machines under three regular objectives. It introduces a swapping operation supported by an additive combinatorics lemma showing that, for any schedule that is locally optimal under the swap, there exists an equivalent optimal schedule where every prefix of jobs has machine loads differing by at most 4p_max squared. This bound lets the algorithm discard many redundant states that differ only in how total processing time is distributed across machines. The resulting algorithms run in O(p_max to the power of 2m minus 2 times n) for minimizing weighted completion time and maximum lateness, and a slightly higher but still improved bound for the weighted number of late jobs.

Core claim

We prove that, whenever this swap does not increase the objective value, there exists an optimal schedule in which, for every prefix of jobs, the load difference between any two machines is at most 4p_max squared. This lets us prune redundant states throughout the dynamic program, replacing the dependence on P by a dependence on p_max squared. We show that the swap is non-increasing for all three objectives above.

What carries the argument

The swapping argument combined with an additive-combinatorial lemma that guarantees the existence of an optimal schedule with bounded prefix load differences.

If this is right

  • Pm||sum w_j C_j and Pm||L_max admit O(p_max^{2m-2} n) algorithms.
  • Pm||sum w_j U_j admits O(p_max^{2m-2} P n) time which is at most O(p_max^{2m-1} n^2).
  • When p_max is polylogarithmic in n the first two problems obtain near-linear time in n.
  • The new bounds are strictly faster than the original Lawler-Moore times whenever p_max is o(sqrt(P)).

Where Pith is reading between the lines

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

  • The same bounded-difference property may allow state pruning in other dynamic programs that track machine loads for regular scheduling objectives.
  • If similar swapping arguments apply to unrelated-machine or precedence-constrained settings, dependence on total processing time could be reduced there as well.
  • When all processing times are bounded by a small constant the new algorithms become polynomial in n with degree independent of m for the first two objectives.

Load-bearing premise

The swap operation never increases the objective value and the combinatorial lemma accurately limits how much machine loads can differ in some optimal schedule.

What would settle it

An instance and objective where every optimal schedule has at least one job prefix with two machines whose loads differ by more than 4p_max squared would show that the pruning step discards necessary states.

Figures

Figures reproduced from arXiv: 2604.13642 by Danny Hermelin, Dvir Shabtay, Karl Bringmann, Tomohiro Koana.

Figure 1
Figure 1. Figure 1: Gantt chart depicting machine loads. Light gray blocks represent the jobs {1, . . . , j} already considered in the dynamic programming state, while dark gray blocks represent future jobs in the priority sequence. The state space is pruned by ensuring the load difference between any two machines at step j remains within O(p 2 max). We use this additive combinatorics lemma to prove that there exists an optim… view at source ↗
Figure 2
Figure 2. Figure 2: A depiction of the intermediate schedule σ ′ obtained from a swap performed between machines Mh and Mi. The lighter gray depicts jobs in Jj = {1, . . . , j}. The darker gray depicts jobs in J \ Jj . In the final schedule σ ′′, the jobs of JH′ and JI ′′ can move to the left and right on Mi, while the jobs of JI ′ can move to the right on Mh. Suppose these three conditions hold for σ. Let ti := P(Ji,j (σ)) d… view at source ↗
read the original abstract

The Lawler-Moore dynamic programming framework is a classical tool in scheduling on parallel machines. It applies when the objective is regular, i.e. monotone in job completion times, and each machine follows a fixed priority order such as Smith's Rule or Jackson's Rule. For the basic objectives $Pm||\sum w_jC_j$, $Pm||L_{\max}$, and $Pm||\sum w_jU_j$, it gives running times $O(P^{m-1}n)$, $O(P^{m-1}n)$, and $O(P^mn)$, respectively, where $P$ is the total processing time. Recent SETH-based lower bounds indicate that the dependence on $P$ is essentially optimal, but they do not rule out improved dependence on the maximum processing time $p_{\max}$. We give the first major speedup of the Lawler-Moore recurrence. Our main ingredients are a new state-pruning method and a swapping argument based on an additive-combinatorial lemma. We prove that, whenever this swap does not increase the objective value, there exists an optimal schedule in which, for every prefix of jobs, the load difference between any two machines is at most $4p_{\max}^2$. This lets us prune redundant states throughout the dynamic program, replacing the dependence on $P$ by a dependence on $p_{\max}^2$. We show that the swap is non-increasing for all three objectives above. Hence $Pm||\sum w_jC_j$ and $Pm||L_{\max}$ admit algorithms with running time $O(p_{\max}^{2m-2}n)$, while $Pm||\sum w_jU_j$ can be solved in time $O(p_{\max}^{2m-2}Pn)\le O(p_{\max}^{2m-1}n^2)$. These bounds strictly improve the original Lawler-Moore runtimes whenever $p_{\max}=o(\sqrt{P})$. In particular, for $Pm||\sum w_jC_j$ and $Pm||L_{\max}$, we obtain the first near-linear-time algorithms when processing times are polylogarithmic in $n$.

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

0 major / 3 minor

Summary. The paper improves the classical Lawler-Moore dynamic programming framework for three regular scheduling objectives on m parallel machines (Pm||∑w_j C_j, Pm||L_max, and Pm||∑w_j U_j). It introduces a state-pruning technique derived from a swapping argument and a new additive-combinatorial lemma, proving that there exists an optimal schedule in which, for every prefix of jobs, the load difference between any two machines is at most 4p_max². This replaces the original O(P^{m-1}n) or O(P^m n) dependence on total processing time P with a dependence on p_max², yielding running times O(p_max^{2m-2}n) for the first two problems and O(p_max^{2m-2} P n) ≤ O(p_max^{2m-1} n²) for the third. The swap is shown to be non-increasing for each objective.

Significance. If the combinatorial lemma and non-increasing property hold, the result is a substantial advance: the first major asymptotic improvement to Lawler-Moore DP in decades, with near-linear time when p_max is polylogarithmic in n. The hereditary prefix property and direct verification for each objective avoid circularity or fitted parameters, and the approach may generalize to other DP-based scheduling problems.

minor comments (3)
  1. Abstract: the inequality O(p_max^{2m-2} P n) ≤ O(p_max^{2m-1} n²) holds when P = O(p_max n), which is always true but should be stated explicitly for clarity.
  2. The manuscript should include a self-contained statement of the additive-combinatorial lemma (including the precise constant 4 and any assumptions on processing times) in the main body, even if the proof is deferred to an appendix.
  3. It would strengthen the presentation to include a brief comparison table of the new versus original Lawler-Moore exponents for small fixed m (e.g., m=2,3).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our paper, as well as the recommendation for minor revision. No specific major comments appear in the report, so we have no points to address individually at this stage. We remain available to incorporate any additional minor suggestions from the editor or referee.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central speedup derives from a new additive-combinatorial lemma and a swapping argument proven directly for the three objectives (sum w_j C_j, L_max, sum w_j U_j). The key existence claim—that an optimal schedule exists with prefix load differences at most 4 p_max^2—is established by verifying that the swap is non-increasing for each objective, then using the hereditary prefix property to prune DP states. No step reduces by construction to fitted parameters, self-citations, or tautological definitions; the reduction from dependence on P to p_max^2 follows from the independent combinatorial bound.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the additive-combinatorics lemma and the proof that the swap is non-increasing for the three objectives. No free parameters are fitted to data, as this is a worst-case algorithmic analysis.

axioms (1)
  • domain assumption Additive combinatorics lemma that enables bounding machine load differences by 4p_max^2
    Invoked to prove existence of an optimal schedule with the stated load property for any prefix.

pith-pipeline@v0.9.0 · 5712 in / 1407 out tokens · 60791 ms · 2026-05-10T12:16:23.346350+00:00 · methodology

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Reference graph

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