Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.27674

Coloring Queens with Thousands of Encodings

Bernardo Subercaseaux·Benjamin Przybocki·Marijn J. H. Heule
View original source ↗Read source document ↗Export metadata ↓

Abstract

In The Art of Computer Programming, Knuth benchmarked 10 encoding techniques for computing the chromatic number of the queen's graph: the minimum number of colors needed to color the squares of an $n \times n$ chessboard so that no two squares sharing a row, column, or diagonal receive the same color. In this paper, we extend his analysis much further by comparing thousands of encodings for the same problem, which allows us to identify additional factors that are important for solver performance. We obtain 1584 encodings for this problem by varying (a) the constraints that encode which color is assigned to each cell, (b) the constraints that forbid the same color appearing in a row, column, or diagonal line, and (c) the symmetry-breaking constraints. We find that the three most impactful encoding factors are (i) the choice of symmetry-breaking constraints, (ii) enabling so-called clique hints, and (iii) enforcing that each cell is assigned exactly one color through blocked clauses. Furthermore, while Knuth proposed clique hints as an advantage of the order encoding, we show in fact that they can be effectively employed for the one-hot encoding as well.

Explore related subjects

cs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bernardo Subercaseaux, Benjamin Przybocki, Marijn J. H. Heule. 2026-09-23. Coloring Queens with Thousands of Encodings. https://arxiv.org/abs/2609.27674

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Completeness of Kozen's Axiomatization for the Modal mu-Calculus: A Simple Proof

The modal mu-calculus, introduced by Dexter Kozen, is an extension of modal logic with fixpoint operators. Its axiomatization, Koz, was introduced at the same time and is an extension of the minimal modal logic K with the so-called Park fixpoint induction principle. It took more than a decade for the completeness of Koz to be proven, finally achieved by Igor Walukiewicz. However, his proof is fairly involved. In this article, we present an improved proof for the completeness of Koz which, although similar to the original, is simpler and easier to understand. Keywords: The modal mu-calculus, completeness, omega-automata.

cs.LO↗
arXiv2026

Blurred Drinker Paradoxes and Blurred Choice Axioms: Constructive Reverse Mathematics of the Downward Löwenheim-Skolem Theorem

In the setting of constructive reverse mathematics, we analyse the downward Löwenheim-Skolem (DLS) theorem of first-order logic, stating that every infinite model has a countable elementary submodel. Refining the well-known equivalence of the DLS theorem to the axiom of dependent choice (DC) over classical base theories, our constructive approach allows for several finer logical decompositions: Just assuming countable choice (CC), the DLS theorem is equivalent to the conjunction of DC with a newly identified fragment of the excluded middle (LEM) that we call the blurred drinker paradox (BDP). Further without CC, the DLS theorem is equivalent to the conjunction of BDP with similarly blurred weakenings of DC and CC. Independently of their connection with the DLS theorem, we also study BDP and the blurred choice axioms on their own, for instance by showing that BDP is LEM without a contribution of Markov's principle and that blurred DC is DC without a contribution of CC. The paper is hyperlinked with an accompanying Coq development.

cs.LO↗
arXiv2026

Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems

We establish mathematical limits of algorithmic safety verification for Turing-complete self-modifying systems, the class in which recursive self-improvement takes place, both for a fixed system and across its own modification. Statically, no verifier is sound, complete and tractable: over unbounded domains by Rice's and Gödel's theorems, over all finite configurations by Trakhtenbrot's theorem, and over succinctly described finite environments because verifying a policy against an adversary is coNP-complete and synthesising one is PSPACE-complete. Dynamically, we model one step of self-modification as a computable transformation of code and ask whether a safety property survives it. If the transformation depends only on behaviour, this is Rice's theorem one level up; if it reads the code, as self-modification does, the question is no longer semantic, yet the same s-m-n reduction works inside a class of behaviourally identical programs and inherits the halting degree. One step is never harder than the property; persistence along the whole trajectory can be $Π^0_2$-complete. Certification by a total algorithm is possible only for transformations of restricted expressivity, not merely for systems that stop changing. No tower of supervisors helps, and every total supervisor errs on an undecidable set of systems. For effectively pointwise properties, every faithful bounded scheme that certifies on finite behavioural evidence admits evolution traces certified at every stage while the property is violated. What survives is exact: a monitor that raises an alarm on violation semidecides it, and comparison against a frozen reference keeps the full theory.

cs.LO↗