List coloring extends graph coloring by assigning each vertex a list of allowed colors. A graph is k-choosable if it can be properly colored for any choice of lists with k colors each. Deciding k-choosability is π²ₚ-complete, bipartite graphs have unbounded list chromatic number, and planar graphs (famously 4-colorable) are all 5-choosable but not all 4-choosable. To search for graphs of given choosability, we extend SAT Modulo Symmetries (SMS) with custom propagators for list coloring pruning techniques and propose a quantified Boolean (QBF) encoding for choosability. We employ a hybrid approach: pen-and-paper reasoning to optimize our formulas followed by automated case distinction by QBF solvers and SMS. Our methods yield two significant results: (1) a 27-vertex planar graph that is 4-choosable yet cannot be proven so using the combinatorial Nullstellensatz widely applied in previous work (we show this is a smallest graph with that property), and (2) the smallest graph exhibiting a gap between chromatic and list chromatic numbers for chromatic number 3.

Parallel solving via cube-and-conquer is a key method for solving hard instances with SAT. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers augmented with propagators presents unique challenges as propagators learn constraints dynamically during the search. We study this problem using SAT Modulo Symmetries (SMS) as our primary test case. In our setting, the SMS symmetry-breaking propagator is an ordinary IPASIR-UP propagator; the techniques below do not rely on properties specific to symmetry breaking, except in the benchmark instantiations. Through extensive experimentation comprising over 20,000 CPU hours, we systematically evaluate different cube-and-conquer variants on three well-studied combinatorial problems. Our methodology combines prerun phases to collect learned constraints, various cubing strategies, and parameter tuning via algorithm configuration. The comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving. Our best method reduces total solving time by factors of 2-10x from improved cubing, and reduces the time for the hardest cubes by factors of 2-50x.

Abstract In this paper, we present the first proof-of-concept framework for end-to-end formally verified graph generation. Our approach integrates SAT modulo symmetries with the Lean proof assistant, providing a unified, machine-checked verification pipeline that spans high-level graph-theoretic specifications, propositional encodings, symmetry-breaking mechanisms, and solver-based search. By formalizing graph invariance and symmetry reasoning within Lean, we eliminate common trust assumptions and obtain fully certified non-existence results. We evaluate the framework on three benchmark classes of graph-generation problems, showing practical feasibility on nontrivial unsatisfiable instances.

Graph generation and enumeration problems often require handling equivalent graphs---those that differ only in vertex labeling. We study how to extend SAT Modulo Symmetries (SMS), a framework for eliminating such redundant graphs, to handle more complex constraints. While SMS was originally designed for constraints in propositional logic (in NP), we now extend it to handle quantified Boolean formulas (QBF), allowing for more expressive specifications like non-3-colorability (a coNP-complete property). We develop two approaches: a static QBF encoding and a dynamic method integrating SMS into QBF solvers. Our analysis reveals that while specialized approaches can be faster, QBF-based methods offer easier implementation and formal verification capabilities.

Envy-freeness up to any good (EFX) is a key concept in Computational Social Choice for the fair division of indivisible goods, where no agent envies another’s allocation after removing any single item. A deeper understanding of EFX allocations is facilitated by exploring the rainbow cycle number (R_f(d)), the largest number of independent sets in a certain class of directed graphs. Upper bounds on R_f(d) provide guarantees to the feasibility of EFX allocations (Chaudhury et al., EC 2021). In this work, we precisely compute the numbers R_f(d) for small values of d, employing the SAT modulo Symmetries framework (Kirchweger and Szeider, CP 2021). SAT modulo Symmetries is tailored specifically for the constraint-based isomorph-free generation of combinatorial structures. We provide an efficient encoding for the rainbow cycle number, comparing eager and lazy approaches. To cope with the huge search space, we extend the encoding with invariant pruning, a new method that significantly speeds up computation.

Modern SAT solvers are often integrated as sub-reasoning engines into more complex tools to address problems beyond the Boolean satisfiability problem. Consider, for example, solvers for Satisfiability Modulo Theories (SMT), combinatorial optimization, model enumeration, and model counting. There, the SAT solver can often provide relevant information beyond the satisfiability answer and the domain knowledge of the embedding system, such as symmetry properties or theory axioms, may benefit the CDCL search. However, this knowledge can often not be efficiently represented in clausal form. This paper proposes a general interface to inspect and influence the internal behaviour of CDCL SAT solvers. The aim is to capture the essential functionalities that simplify and improve use cases requiring a more fine-grained interaction with the SAT solver than provided via the standard IPASIR interface. For our experiments, the state-of-the-art SAT solver CaDiCaL is extended with the proposed interface and evaluated on two representative use cases: enumerating graphs within the SAT modulo Symmetries framework (SMS), and as the main CDCL(T) SAT engine of the SMT solver cvc5.

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty–Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

We present a new SAT-based method for generating all graphs up to isomorphism that satisfy a given co-NP property. Our method extends the SAT Modulo Symmetry (SMS) framework with a technique that we call co-certificate learning. If SMS generates a candidate graph that violates the given co-NP property, we obtain a certificate for this violation, i.e., `co-certificate' for the co-NP property. The co-certificate gives rise to a clause that the SAT solver, serving as SMS's backend, learns as part of its CDCL procedure. We demonstrate that SMS plus co-certificate learning is a powerful method that allows us to improve the best-known lower bound on the size of Kochen-Specker vector systems, a problem that is central to the foundations of quantum mechanics and has been studied for over half a century. Our approach is orders of magnitude faster and scales significantly better than a recently proposed SAT-based method.

The shortest common supersequence problem (SCSP) is a well-known NP-hard problem with many applications, in particular in data compression, computational molecular biology, and text editing. It aims at finding for a given set of input strings a shortest string such that every string from the set is a subsequence of the computed string. Due to its NP-hardness, many approaches have been proposed to tackle the SCSP heuristically. The currently best-performing one is based on beam search (BS). In this paper, we present a novel heuristic (AEL) for guiding a BS, which approximates the expected length of an SCSP of random strings, and embed the proposed heuristic into a multilevel probabilistic beam search (MPBS). To overcome the arising scalability issue of the guidance heuristic, a cut-off approach is presented that reduces large instances to smaller ones. The proposed approaches are tested on two established sets of benchmark instances. MPBS guided by AEL outperforms the so far leading method on average on a set of real instances. For many instances new best solutions could be obtained.

Rota's basis conjecture (RBC) states that given a collection $\mathcal{B}$ of $n$ bases in a matroid $M$ of rank $n$, one can always find $n$ disjoint rainbow bases with respect to $\mathcal{B}$. In this paper, we show that if $M$ has girth at least $n-o(\sqrt{n})$, and no element of $M$ belongs to more than $o(\sqrt{n})$ bases in $\mathcal{B}$, then one can find at least $n - o(n)$ disjoint rainbow bases with respect to $\mathcal{B}$. This result can be seen as an extension of the work of Geelen and Humphries, who proved RBC in the case where $M$ is paving, and $\mathcal{B}$ is a pairwise disjoint collection. We make extensive use of the cascade idea introduced by Bucić et al.

Answer Set Programming (ASP) is a model, ground, and solve paradigm. The integration of application- or theory-specific reasoning into ASP systems thus impacts on many if not all elements of its workflow, viz. input language, grounding, intermediate language, solving, and output format. We address this challenge with the fifth generation of the ASP system clingo and its grounding and solving components by equipping them with well-defined generic interfaces facilitating the manifold integration efforts. On the grounder's side, we introduce a generic way of specifying language extensions and propose an intermediate format accommodating their ground representation. At the solver end, this is accompanied by high-level interfaces easing the integration of theory propagators dealing with these extensions.