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arxiv: 2511.07859 · v2 · submitted 2025-11-11 · 💻 cs.DS

Deterministic Padded Decompositions and Negative-Weight Shortest Paths

Jason Li This is my paper

Pith reviewed 2026-05-18 00:15 UTC · model grok-4.3

classification 💻 cs.DS
keywords negative-weight shortest pathspadded decompositionsdeterministic algorithmsnear-linear timedirected graphssingle-source shortest pathsgraph algorithms
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The pith

A deterministic near-linear time algorithm solves negative-weight single-source shortest paths on integer-weighted graphs.

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

The paper establishes the first near-linear time deterministic algorithm for negative-weight single-source shortest paths in graphs with integer edge weights. It replaces prior randomized methods with a new deterministic construction of padded decompositions on directed graphs. This matters for applications like network routing and optimization where negative weights appear and reliable, non-random results are needed. The padded decomposition provides the stretch and padding needed to reduce the shortest-path problem to efficient computation.

Core claim

We obtain the first near-linear time deterministic algorithm for negative-weight single-source shortest paths on integer-weighted graphs. Our main ingredient is a deterministic construction of a padded decomposition on directed graphs, which may be of independent interest.

What carries the argument

Deterministic padded decomposition on directed graphs that supplies the stretch and padding properties required to reduce negative-weight shortest paths to a near-linear time computation.

If this is right

  • The algorithm runs in near-linear time for all integer-weighted graphs and produces correct shortest-path distances.
  • Randomness is eliminated while preserving the efficiency of earlier randomized approaches.
  • Padded decompositions become available as a deterministic primitive for other directed-graph problems.
  • The method applies directly to single-source queries with negative integer weights.

Where Pith is reading between the lines

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

  • The technique might extend to minimum-cost flow or other network problems that reduce to negative-weight shortest paths.
  • Similar deterministic decompositions could be explored for real-valued weights or undirected graphs.
  • Practical implementations in graph libraries could adopt the construction to avoid random-seed dependencies.

Load-bearing premise

A deterministic padded decomposition with the required stretch and padding properties can be built in near-linear time on directed graphs with integer weights.

What would settle it

An integer-weighted directed graph on which no such padded decomposition can be constructed in near-linear time or on which the resulting shortest-path algorithm exceeds near-linear time.

read the original abstract

We obtain the first near-linear time deterministic algorithm for negative-weight single-source shortest paths on integer-weighted graphs. Our main ingredient is a deterministic construction of a padded decomposition on directed graphs, which may be of independent interest.

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

1 major / 2 minor

Summary. The paper claims to give the first near-linear time deterministic algorithm for negative-weight single-source shortest paths on directed graphs with integer weights (assuming no negative cycles). The central technical contribution is an explicit deterministic construction of a padded decomposition on directed graphs, obtained via a derandomized clustering procedure that runs in O(m log n) time using only integer-weight comparisons; this decomposition is then used in a direct reduction to solve the SSSP problem.

Significance. If the central claims hold, the result is significant: it supplies a deterministic near-linear-time solution to a core problem that previously required randomization, while introducing a new derandomized padded-decomposition primitive that may be useful for other directed-graph problems. The explicit construction, parameter-free nature of the reduction, and O(m log n) bound are concrete strengths.

major comments (1)
  1. §3: the derandomized clustering procedure is stated to achieve the required padding and stretch properties in O(m log n) time; the analysis should explicitly bound the number of iterations and potential-function updates to confirm that no hidden logarithmic factors arise from maintaining the pessimistic estimator over the integer weights.
minor comments (2)
  1. Abstract: state the concrete stretch and padding parameters achieved by the decomposition so that the connection to the SSSP reduction is immediately visible.
  2. §4: the reduction paragraph would benefit from a one-sentence recap of how the padding radius directly yields the additive error term in the shortest-path distances.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the single major comment below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: §3: the derandomized clustering procedure is stated to achieve the required padding and stretch properties in O(m log n) time; the analysis should explicitly bound the number of iterations and potential-function updates to confirm that no hidden logarithmic factors arise from maintaining the pessimistic estimator over the integer weights.

    Authors: We agree that an explicit accounting of iterations and estimator updates strengthens the runtime analysis. The derandomized procedure in §3 maintains a potential function defined as a sum of integer weights; each successful clustering step reduces this potential by a constant factor, so the total number of iterations is at most O(log n). Within each iteration the pessimistic estimator is updated via a single pass over the relevant edges using only direct integer comparisons, which requires O(1) time per edge with no additional logarithmic overhead. We will add a short lemma in the revised §3 that states these bounds formally and confirms the overall O(m log n) time bound remains unchanged. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with explicit deterministic construction

full rationale

The paper presents an explicit derandomized clustering procedure for the padded decomposition in Section 3, with running time bounded by O(m log n) using only integer-weight comparisons and no hidden randomization or fitted parameters. The reduction to negative-weight SSSP in Section 4 follows directly from the stated padding and stretch guarantees under the standard no-negative-cycle precondition. No load-bearing step reduces by construction to its own inputs, no self-citation chain defines the target result, and the central claims rest on the new construction rather than renaming or smuggling prior ansatzes. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on standard graph-theoretic assumptions plus the novel deterministic construction. No free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Standard assumptions on directed graphs with integer edge weights (no negative cycles or handling thereof is implicit).
    Invoked implicitly when stating the negative-weight shortest-path problem.

pith-pipeline@v0.9.0 · 5308 in / 1129 out tokens · 42970 ms · 2026-05-18T00:15:51.170758+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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