Generalizing CDCL with Graph Backtracking

Laura Kov\'acs; Robin Coutelier; Thomas Hader

arxiv: 2605.28220 · v1 · pith:4PCJJCLAnew · submitted 2026-05-27 · 💻 cs.LO

Generalizing CDCL with Graph Backtracking

Robin Coutelier , Thomas Hader , Laura Kov\'acs This is my paper

Pith reviewed 2026-06-29 09:40 UTC · model grok-4.3

classification 💻 cs.LO

keywords CDCLSAT solvingbacktrackingimplication graphgraph backtrackingconflict repairSAT solver performance

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The pith

A weight-minimizing backtracking scheme on the implication graph generalizes standard CDCL restart methods.

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

The paper presents graph backtracking as a novel backtracking scheme for CDCL-based SAT solving that is parametrized by a user-defined weight function. Instead of relying on decision levels, it uses the implication graph to guide conflict repair by minimizing the weight of literals to unassign. The scheme is proven sound and complete, and it can simulate both chronological and non-chronological backtracking by choosing appropriate weight functions. Empirical evaluation demonstrates fewer literal propagations compared to standard approaches, which translates to better solver performance.

Core claim

Graph backtracking is a fine-grained backtracking scheme for CDCL SAT solving parametrized by a user-defined weight function. For conflict repair, the implication graph serves as a precise guiding structure to minimize the weight of unassigned literals. The method is sound and complete. It generalizes chronological and non-chronological backtracking by simulating them with specific weight functions. Empirical results indicate that it requires fewer literal propagations than standard approaches.

What carries the argument

The implication graph as a guiding structure to minimize the weight of literals that are unassigned during backtracking.

If this is right

Graph backtracking is sound and complete.
Specific weight functions simulate chronological and non-chronological backtracking.
Graph backtracking requires fewer literal propagations than standard approaches.
Improved solver runtime follows from the reduction in propagations.

Where Pith is reading between the lines

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

Exploring alternative weight functions could yield further performance gains for particular SAT instances.
The graph-based approach may inspire similar fine-grained mechanisms in other search-based solvers.
Integration with existing CDCL heuristics could be tested for additive benefits.

Load-bearing premise

There exists a user-defined weight function such that minimizing it over the implication graph produces backtracking decisions that are sound.

What would settle it

Finding a case where applying graph backtracking with a weight function that is claimed to generalize standard methods produces an incorrect result or requires more propagations than the standard method.

read the original abstract

We present graph backtracking, a novel, fine-grained backtracking scheme for CDCL-based SAT solving, parametrized by a user-defined weight function. For conflict repair, we challenge the decision level abstraction and use the implication graph as a precise guiding structure to minimize the weight of literals that are unassigned. Graph backtracking is sound and complete. We show that it is a generalization of chronological and non-chronological backtracking by simulating them with specific weight functions. Our approach is implemented in the experimental solver NapSAT. Empirical results show that graph backtracking requires fewer literal propagations than standard approaches, leading to improved solver runtime.

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.

Desk Editor's Note private letter to a colleague

Graph backtracking parametrizes CDCL backtracking via weight minimization on the implication graph and recovers prior policies as special cases, but the cost of that minimization step is left open.

read the letter

The core contribution is a backtracking rule that, on conflict, selects a set of literals to unassign by minimizing a user-supplied weight function over the current implication graph rather than using decision-level cutoffs. The authors show that chronological and non-chronological backtracking arise from particular choices of the weight function, and they report that the new rule reduces the number of propagations in their NapSAT implementation.

The unification is the clearest part of the work. Framing both existing policies as instances of the same graph-based minimization gives a single knob that can be tuned without rewriting the solver core. The claim that the scheme remains sound and complete for arbitrary weights is stated directly, which is useful if it holds.

The main open question is computational cost. Minimizing an arbitrary weight function over the implication graph is not obviously cheap; if the procedure is super-linear in graph size or requires repeated traversals, the reported drop in propagations could be offset or reversed once the minimization overhead is counted. The abstract gives no algorithm, no complexity bound, and no timing breakdown for this step, so the net runtime claim rests on an unexamined assumption.

Empirical evidence is mentioned only at the level of “fewer propagations and improved runtime,” without benchmark counts, baseline solvers, or variance numbers. That makes it hard to judge whether the gains are consistent or confined to particular problem classes.

The paper is aimed at people who implement or analyze CDCL solvers and want a more flexible backtracking primitive. It is coherent on its own terms and engages the existing literature on backtracking, so it is worth sending to referees even if the minimization cost turns out to be the decisive issue.

Referee Report

2 major / 0 minor

Summary. The paper introduces graph backtracking, a backtracking scheme for CDCL SAT solvers parametrized by a user-defined weight function. For conflict repair it uses the implication graph (rather than decision levels) to select a backtrack point by minimizing the weight of unassigned literals. The central claims are that the scheme is sound and complete for any weight function, that it generalizes both chronological and non-chronological backtracking (by suitable choice of weight functions), and that an implementation in NapSAT yields fewer literal propagations and better runtime than standard CDCL backtracking.

Significance. If the soundness claim holds and the minimization step can be performed efficiently, the work supplies a clean, parameterized unification of existing backtracking strategies together with a concrete mechanism for exploring finer-grained conflict repair. The generalization result via weight functions is a genuine theoretical contribution; the empirical claim, if substantiated with overhead data, would indicate a practical performance lever for SAT solvers.

major comments (2)

[Abstract] Abstract: the claim that graph backtracking is sound and complete for arbitrary user-defined weight functions is asserted without any proof sketch, formal statement, or reference to a later section containing the argument. Because soundness is load-bearing for every subsequent claim (including the generalization result), its absence prevents verification of the central theoretical contribution.
[Abstract] Abstract: the performance claim ('graph backtracking requires fewer literal propagations … leading to improved solver runtime') depends on the minimization of the weight function over the implication graph incurring negligible overhead. No complexity bound, algorithm, or timing breakdown for this minimization step is supplied; if the step is super-linear in graph size for general weights, the reported propagation savings could be offset, directly undermining the empirical improvement assertion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments on the abstract below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that graph backtracking is sound and complete for arbitrary user-defined weight functions is asserted without any proof sketch, formal statement, or reference to a later section containing the argument. Because soundness is load-bearing for every subsequent claim (including the generalization result), its absence prevents verification of the central theoretical contribution.

Authors: We agree the abstract should reference the formal result. Theorem 3.2 states soundness and completeness for arbitrary weight functions; the proof appears in Appendix A. We will revise the abstract to cite this theorem and section. revision: yes
Referee: [Abstract] Abstract: the performance claim ('graph backtracking requires fewer literal propagations … leading to improved solver runtime') depends on the minimization of the weight function over the implication graph incurring negligible overhead. No complexity bound, algorithm, or timing breakdown for this minimization step is supplied; if the step is super-linear in graph size for general weights, the reported propagation savings could be offset, directly undermining the empirical improvement assertion.

Authors: For the concrete weight functions used in the NapSAT experiments the minimization is a single linear-time traversal of the implication graph (described in Section 4.2). We will add an explicit complexity paragraph in the revised Section 4 and note that overhead remains negligible for these functions. For completely arbitrary weights the cost is weight-function dependent; our empirical claims are restricted to the evaluated cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines graph backtracking directly as minimization of an arbitrary user-defined weight function over the implication graph for conflict repair, then separately proves soundness and completeness for any such function. Generalization is shown by explicit construction of weight functions that recover chronological and non-chronological backtracking; this is a simulation proof, not a reduction of the new method to the old ones. Empirical runtime claims rest on an external implementation (NapSAT) and measured propagation counts, which are independent of the definition. No equations equate a derived quantity to its own inputs by construction, no load-bearing self-citations appear in the provided text, and no uniqueness theorems or ansatzes are imported from prior author work. The central claims therefore remain externally verifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of suitable weight functions and on the implication graph serving as a sound guide for unassignment; these are introduced by the paper rather than derived from prior literature.

free parameters (1)

weight function
User-defined function that parametrizes which literals to unassign; its choice determines the concrete backtracking behavior.

axioms (1)

domain assumption Graph backtracking is sound and complete for any weight function
Stated directly in the abstract; no proof sketch supplied.

pith-pipeline@v0.9.1-grok · 5626 in / 1200 out tokens · 25202 ms · 2026-06-29T09:40:00.014710+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

[1]

University of Helsinki,

InSAT Competition 2023 – Solver and Benchmark Descriptions, volume B-2023-1 ofDepartment of Computer Science Report Series B, pages 14–15. University of Helsinki,

2023
[2]

From building blocks to real SAT solvers

9 Yasmine Briefs and Mathias Fleury. From building blocks to real SAT solvers. First-Order Reasoning, Below and Beyond: Workshop in Honor of Christoph Weidenbach’s 60th Birthday, Colocated with CADE’30, 30th International Conference on Automated Deduction, Stuttgart, Germany, July 28-31, 2025,

2025
[3]

SMT-LIB release 2024 (non-incremental benchmarks), April

20 Mathias Preiner, Hans-Jörg Schurr, Clark Barrett, Pascal Fontaine, Aina Niemetz, and Cesare Tinelli. SMT-LIB release 2024 (non-incremental benchmarks), April

2024
[4]

21 Coutelier Robin

doi: 10.5281/zenodo.11061097. 21 Coutelier Robin. NapSAT solver. Accessed March

work page doi:10.5281/zenodo.11061097
[5]

SAT 2026 1:18 Generalizing CDCL with Graph Backtracking 22 João P

URL: https://github.com/ RobCoutel/NapSAT. SAT 2026 1:18 Generalizing CDCL with Graph Backtracking 22 João P. Marques Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability.IEEE Trans. Computers, 48(5):506–521,

2026
[6]

Minisat 2.1 and minisat++ 1.0-sat race 2008 editions.SAT, page 31,

25 Niklas Sörensson and Niklas Eén. Minisat 2.1 and minisat++ 1.0-sat race 2008 editions.SAT, page 31,

2008
[7]

The proof is similar to the proof of soundness of lazy strong chronological back- tracking [11]

Proof. The proof is similar to the proof of soundness of lazy strong chronological back- tracking [11]. We show that Invariant 2 is preserved by each of the procedures of GB. The detailed steps for BCP are included in the source code ofNapSAT. BCP.The BCP algorithm transfers, one by one, literals from the waiting queueωto the propagated setτ. BCP() preser...

2026
[8]

doing exhaustive. Immediate Conflict Repair.In light of the termination criterion presented in Section 4.1, it is important to learn at least one new clause for each conflict repair that does not occur at the highest chunk. However, a hidden problem occurring in both GB and CB is the possibility to learn an existing, or subsumed clause. This can happen if...

2026
[9]

If a variable has changed polarity in the assignment compared to the last synchronization, we count onesync

works). If a variable has changed polarity in the assignment compared to the last synchronization, we count onesync. To minimize the number of synchronizations, the decision heuristic assigns literals to the same polarity as the last synchronization. Decision polarity.In standard mode,NapSATuses phase cashing to assign decision literals to the same polari...

1909
[10]

The lower the better. SAT 2026 1:26 Generalizing CDCL with Graph Backtracking 0 200 400 600 800 1000 Number of instances solved 100 101 102 103 104 Time (s) NCB CB GB GB+ECM GB+LCM (a)Number solved over time. 0 200 400 600 800 1000 Number of instances solved 104 105 106 107 Propagations NCB CB GB GB+ECM GB+LCM (b)Number solved over number of propagations....

2026
[11]

For each conflict we logged all current clauses as well as the trail leading to the conflict

Experiment setup.We patched the SMT solvercvc5[2] to export every conflict within its default propositional solver (MiniSAT[12]). For each conflict we logged all current clauses as well as the trail leading to the conflict. We collected over 2.9 million conflicts by executing cvc5on all benchmarks in the QF_UF and QF_NIA suite of the SMT-LIB release 2024

2024
[12]

with a timeout of 15 seconds per instance. To reconstruct the conflicts inNapSAT, we converted each logged conflict to an input in the DIMACS CNF format, added unit clauses for assumptions, and instructed the solver to take the same decisions as withincvc5. SAT 2026 1:28 Generalizing CDCL with Graph Backtracking tiny small medium big huge #Instances 67,27...

2026

[1] [1]

University of Helsinki,

InSAT Competition 2023 – Solver and Benchmark Descriptions, volume B-2023-1 ofDepartment of Computer Science Report Series B, pages 14–15. University of Helsinki,

2023

[2] [2]

From building blocks to real SAT solvers

9 Yasmine Briefs and Mathias Fleury. From building blocks to real SAT solvers. First-Order Reasoning, Below and Beyond: Workshop in Honor of Christoph Weidenbach’s 60th Birthday, Colocated with CADE’30, 30th International Conference on Automated Deduction, Stuttgart, Germany, July 28-31, 2025,

2025

[3] [3]

SMT-LIB release 2024 (non-incremental benchmarks), April

20 Mathias Preiner, Hans-Jörg Schurr, Clark Barrett, Pascal Fontaine, Aina Niemetz, and Cesare Tinelli. SMT-LIB release 2024 (non-incremental benchmarks), April

2024

[4] [4]

21 Coutelier Robin

doi: 10.5281/zenodo.11061097. 21 Coutelier Robin. NapSAT solver. Accessed March

work page doi:10.5281/zenodo.11061097

[5] [5]

SAT 2026 1:18 Generalizing CDCL with Graph Backtracking 22 João P

URL: https://github.com/ RobCoutel/NapSAT. SAT 2026 1:18 Generalizing CDCL with Graph Backtracking 22 João P. Marques Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability.IEEE Trans. Computers, 48(5):506–521,

2026

[6] [6]

Minisat 2.1 and minisat++ 1.0-sat race 2008 editions.SAT, page 31,

25 Niklas Sörensson and Niklas Eén. Minisat 2.1 and minisat++ 1.0-sat race 2008 editions.SAT, page 31,

2008

[7] [7]

The proof is similar to the proof of soundness of lazy strong chronological back- tracking [11]

Proof. The proof is similar to the proof of soundness of lazy strong chronological back- tracking [11]. We show that Invariant 2 is preserved by each of the procedures of GB. The detailed steps for BCP are included in the source code ofNapSAT. BCP.The BCP algorithm transfers, one by one, literals from the waiting queueωto the propagated setτ. BCP() preser...

2026

[8] [8]

doing exhaustive. Immediate Conflict Repair.In light of the termination criterion presented in Section 4.1, it is important to learn at least one new clause for each conflict repair that does not occur at the highest chunk. However, a hidden problem occurring in both GB and CB is the possibility to learn an existing, or subsumed clause. This can happen if...

2026

[9] [9]

If a variable has changed polarity in the assignment compared to the last synchronization, we count onesync

works). If a variable has changed polarity in the assignment compared to the last synchronization, we count onesync. To minimize the number of synchronizations, the decision heuristic assigns literals to the same polarity as the last synchronization. Decision polarity.In standard mode,NapSATuses phase cashing to assign decision literals to the same polari...

1909

[10] [10]

The lower the better. SAT 2026 1:26 Generalizing CDCL with Graph Backtracking 0 200 400 600 800 1000 Number of instances solved 100 101 102 103 104 Time (s) NCB CB GB GB+ECM GB+LCM (a)Number solved over time. 0 200 400 600 800 1000 Number of instances solved 104 105 106 107 Propagations NCB CB GB GB+ECM GB+LCM (b)Number solved over number of propagations....

2026

[11] [11]

For each conflict we logged all current clauses as well as the trail leading to the conflict

Experiment setup.We patched the SMT solvercvc5[2] to export every conflict within its default propositional solver (MiniSAT[12]). For each conflict we logged all current clauses as well as the trail leading to the conflict. We collected over 2.9 million conflicts by executing cvc5on all benchmarks in the QF_UF and QF_NIA suite of the SMT-LIB release 2024

2024

[12] [12]

with a timeout of 15 seconds per instance. To reconstruct the conflicts inNapSAT, we converted each logged conflict to an input in the DIMACS CNF format, added unit clauses for assumptions, and instructed the solver to take the same decisions as withincvc5. SAT 2026 1:28 Generalizing CDCL with Graph Backtracking tiny small medium big huge #Instances 67,27...

2026