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arxiv: 1907.00929 · v2 · pith:GXGKKEKHnew · submitted 2019-07-01 · 💻 cs.LO · math.CO

Trimming Graphs Using Clausal Proof Optimization

Marijn J.H. Heule This is my paper

Pith reviewed 2026-05-25 11:16 UTC · model grok-4.3

classification 💻 cs.LO math.CO
keywords minimal unsatisfiable coreproof minimizationSAT solvingunit-distance graphchromatic numberclausal proof optimizationunsatisfiability core
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The pith

Postponing clause deletions during unsatisfiability proof minimization produces smaller minimal unsatisfiable cores than prior methods.

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

The paper presents a technique for computing unsatisfiable cores by gradually minimizing proofs of unsatisfiability while keeping arbitrary clauses in the formula for as long as possible. This contrasts with existing minimal unsatisfiable core algorithms that delete clauses earlier. The method targets cores that are relatively small among all possible minimal cores for a given formula. A reader would care because such cores can reduce the size of structures like graphs that encode hard combinatorial problems. The approach is shown to shrink a known unit-distance graph with chromatic number 5 from 553 vertices to 529 vertices.

Core claim

By postponing deletion of arbitrary clauses from the formula as long as possible, the method computes a minimal unsatisfiable core that is relatively small compared to cores found by existing algorithms. This is achieved through clausal proof optimization applied to propositional formulas.

What carries the argument

The gradual minimization of unsatisfiability proofs by delaying removal of clauses from the input formula.

If this is right

  • The unit-distance graph with chromatic number 5 is reduced to 529 vertices and 2670 edges.
  • The same postponed-deletion strategy can be used to trim other graphs or formulas encoded as SAT problems.
  • Proof minimization becomes a tool for finding compact explanations of unsatisfiability in combinatorial search.

Where Pith is reading between the lines

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

  • The technique may combine with other SAT preprocessing steps to further shrink cores in large-scale verification tasks.
  • Smaller cores could improve the efficiency of proof-based debugging in software or hardware verification.
  • The method suggests a general principle that order of clause removal affects core size, which could be tested on random 3-SAT instances.

Load-bearing premise

Postponing deletion of arbitrary clauses from the formula as long as possible produces a minimal unsatisfiable core that is relatively small compared to cores found by existing algorithms.

What would settle it

An existing minimal unsatisfiable core algorithm applied to the same input formula produces a core smaller than the one obtained by this postponed-deletion method.

Figures

Figures reproduced from arXiv: 1907.00929 by Marijn J.H. Heule.

Figure 1
Figure 1. Figure 1: A justification of the proof of the example formula. Each clause in the proof depends on its incoming arcs. The clauses without incoming arcs represent the formula. In practice, clausal proofs also contain deletion information. The presence of deletion information significantly reduces the cost to compute a justification. Clausal proofs, which can be validated using the RUP method and include dele￾tion inf… view at source ↗
Figure 2
Figure 2. Figure 2: From left to right: illustrations of unit-distance graphs A, B, A ⊕ B, and the Moser Spindle. The graphs shown have chromatic number 2, 2, 3, and 4, respectively. The illustrations show valid colorings with the fewest number of colors. We will use three operations to construct larger and larger graphs: the Minkowski sum [12], rotation, and merging. Given two sets of points A and B, the Minkowski sum of A a… view at source ↗
Figure 3
Figure 3. Figure 3: shows the pseudo code of that algorithm. The procedure RemoveRe￾dundancy removes from a given clausal proof P and justification J all the clauses in P that do not occur in any of the justifications of J. Given a justification J, the procedure ShuffleProof produces a random permutation of the clauses in J such that each clause C appears 1) later in the proof than all the clauses in the justification of C an… view at source ↗
Figure 4
Figure 4. Figure 4: shows the pseudo codes of two algorithms to trim a formula: one algorithm, called TrimFormulaPlain, that simply adds proof optimization to the trimming loop and another one, called TrimFormulaInteract, that additionally interacts with the original formula to further optimize the proof. We focus on the latter algorithm, which is one of the main contributions of this paper. TrimFormulaPlain (formula F) 1 Fco… view at source ↗
Figure 5
Figure 5. Figure 5: A 3-coloring of the graph H1 3 ⊕ H1 3 ⊕ H1 3 (left) and a 4-coloring of the graph H1 3 ⊕ H1 3 ⊕ H1 3 ⊕ H0√ 3+√ 11 6 (right). We observed some interesting patterns when combining the graphs H1 3 and H0√ 3+√ 11 6 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A 4-coloring of the graph H1 3 ⊕ H1 3 ⊕ H1 3 ⊕ H0√ 3+√ 11 6 ⊕H0√ 3+√ 11 6 . H0√ 3+√ 11 6 . Now G2167 equals C13 ⊕ C13 ⊕ C13 ⊕ C13 ⊕ C13 ⊕ C13 ⊕ C13 ⊕ C13 without the points that have a distance larger than 2 from the central vertex. This graph has 2 167 vertices and 16 512 edges and is shown in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A 4-coloring of the graph H1 3 ⊕ H1 3 ⊕ H1 3 ⊕ H0√ 3+√ 11 6 ⊕H0√ 3+√ 11 6 ⊕H0√ 3+√ 11 6 . obvious way to add such points in a way that the resulting graph has chromatic number 5. Another pattern that can be observed in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A 4-coloring of graph G2167. 6.1 Encoding We can compute the chromatic number of a graph G as follows. Construct two formulas, one asking whether G can be colored with k−1 colors, and one whether G can be colored with k colors. Now, G has chromatic number k if and only if the former is unsatisfiable while the latter is satisfiable. The construction of these two formulas can be achieved using the following … view at source ↗
Figure 9
Figure 9. Figure 9: The size (number of clauses) of the unsatisfiable core and the optimized proof of unsatisfiability (y-axis) of the first twenty steps (x-axis) of the OptimizeProof algorithm, when starting with F + 4 and the smallest proof (left) or the largest proof (right). The first type of clauses, called vertex clauses, ensures that each vertex has at least one color, while the second type of clauses, called edge clau… view at source ↗
Figure 10
Figure 10. Figure 10: The size of subgraphs corresponding to the unsatisfiable cores when using the algorithms TrimFormulaPlain (left) and TrimFormulaInteract (right) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A 529-vertex unit-distance graph with chromatic number 5. In the shown coloring, only the origin has the fifth color (white). By combining the new algorithm and the new graph, we were able to reduce the smallest known unit-distance graph with chromatic number 5 to a graph with 529 vertices and 2670 edges (down from 553 vertices and 2720 edges). This graph is also much more symmetric. It is generally easie… view at source ↗
read the original abstract

We present a method to gradually compute a smaller and smaller unsatisfiable core of a propositional formula by minimizing proofs of unsatisfiability. The goal is to compute a minimal unsatisfiable core that is relatively small compared to other minimal unsatisfiable cores of the same formula. We try to achieve this goal by postponing deletion of arbitrary clauses from the formula as long as possible---in contrast to existing minimal unsatisfiable core algorithms. We applied this method to reduce the smallest known unit-distance graph with chromatic number 5 from 553 vertices and 2720 edges to 529 vertices and 2670 edges.

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 / 2 minor

Summary. The paper introduces a proof-minimization technique for computing unsatisfiable cores of propositional formulas that postpones arbitrary clause deletions to obtain MUCs that are smaller than those produced by standard algorithms. The method is applied to the SAT encoding of the chromatic number of a unit-distance graph, trimming the smallest known 5-chromatic example from 553 vertices and 2720 edges to 529 vertices and 2670 edges.

Significance. If the postponing heuristic can be shown to systematically yield smaller MUCs, the approach would strengthen tools for core extraction and minimal counterexample search in SAT-based graph theory. The reported trimming of the de Grey graph constitutes a concrete, falsifiable outcome that could be independently verified; however, the absence of head-to-head size and runtime data against existing MUC extractors leaves the attribution of the reduction to the new strategy unestablished.

major comments (2)
  1. [Abstract] Abstract (and §3 on the goal): the claim that the postponing strategy produces a MUC 'relatively small compared to other minimal unsatisfiable cores' and 'in contrast to existing minimal unsatisfiable core algorithms' is unsupported by any size, runtime, or quality metrics from prior MUC extractors on the same chromatic-number encoding or on standard benchmark suites.
  2. [Application] Application section (graph trimming result): the reduction from 553/2720 to 529/2670 is presented as evidence of the method's effectiveness, yet no ablation or comparison isolates the contribution of postponed deletion versus the underlying proof minimizer or instance-specific properties.
minor comments (2)
  1. [Method] Notation for the core-extraction procedure could be clarified with a short pseudocode listing or explicit invariants.
  2. [Application] The manuscript would benefit from a brief discussion of the computational resources used for the reported run.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the need for supporting data and clearer attribution. We address the two major comments point by point below, agreeing where the manuscript lacks evidence and proposing targeted revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and §3 on the goal): the claim that the postponing strategy produces a MUC 'relatively small compared to other minimal unsatisfiable cores' and 'in contrast to existing minimal unsatisfiable core algorithms' is unsupported by any size, runtime, or quality metrics from prior MUC extractors on the same chromatic-number encoding or on standard benchmark suites.

    Authors: We agree the manuscript provides no head-to-head metrics against prior MUC extractors. The phrase 'in contrast to existing minimal unsatisfiable core algorithms' describes the methodological choice to postpone arbitrary deletions rather than an empirical performance claim. The goal of obtaining a 'relatively small' MUC is not backed by comparative data in the current text. We will revise the abstract and §3 to remove the unsupported 'relatively small' phrasing and clarify that the contrast is algorithmic. revision: yes

  2. Referee: [Application] Application section (graph trimming result): the reduction from 553/2720 to 529/2670 is presented as evidence of the method's effectiveness, yet no ablation or comparison isolates the contribution of postponed deletion versus the underlying proof minimizer or instance-specific properties.

    Authors: The reduction yields a verifiable smaller 5-chromatic unit-distance graph. However, the manuscript contains no ablation experiments separating the postponing heuristic from the base proof minimizer. We will revise the application section to explicitly note this limitation and state that the result demonstrates utility on this instance without claiming isolated attribution to the new strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: concrete computational trimming result with no self-referential derivation

full rationale

The paper describes an algorithmic heuristic (postponing arbitrary clause deletion during proof minimization) whose output is a concrete, instance-specific unsatisfiable core that trims a known graph. No equations, fitted parameters, or claimed first-principles derivations appear; the reported vertex/edge counts are direct results of running the procedure on the chromatic-number encoding rather than quantities forced by definition or by a self-citation chain. The contrast to prior MUC methods is stated as motivation but does not constitute a load-bearing self-referential step.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5619 in / 946 out tokens · 20146 ms · 2026-05-25T11:16:51.942272+00:00 · methodology

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Reference graph

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