Author name cluster

Markus Kirchweger

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8 papers

2 author rows

AAAI Conference 2026 Conference Paper

Graph Choosability via SAT: Beyond the Nullstellensatz

Markus Kirchweger
Tomáš Peitl
David Seka
Stefan Szeider

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.

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AAAI Conference 2025 Conference Paper

Breaking Symmetries in Quantified Graph Search: A Comparative Study

Mikoláš Janota
Markus Kirchweger
Tomáš Peitl
Stefan Szeider

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.

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JAIR Journal 2024 Journal Article

Satisfiability Modulo User Propagators

Katalin Fazekas
Aina Niemetz
Mathias Preiner
Markus Kirchweger
Stefan Szeider
Armin Biere

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.

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SAT Conference 2023 Conference Paper

A SAT Solver's Opinion on the Erdős-Faber-Lovász Conjecture

Markus Kirchweger
Tomás Peitl
Stefan Szeider

In 1972, Paul Erdős, Vance Faber, and Lászlo Lovász asked whether every linear hypergraph with n vertices can be edge-colored with n colors, a statement that has come to be known as the EFL conjecture. Erdős himself considered the conjecture as one of his three favorite open problems, and offered increasing money prizes for its solution on several occasions. A proof of the conjecture was recently announced, for all but a finite number of hypergraphs. In this paper we look at some of the cases not covered by this proof. We use SAT solvers, and in particular the SAT Modulo Symmetries (SMS) framework, to generate non-colorable linear hypergraphs with a fixed number of vertices and hyperedges modulo isomorphisms. Since hypergraph colorability is NP-hard, we cannot directly express in a propositional formula that we want only non-colorable hypergraphs. Instead, we use one SAT (SMS) solver to generate candidate hypergraphs modulo isomorphisms, and another to reject them by finding a coloring. Each successive candidate is required to defeat all previous colorings, whereby we avoid having to generate and test all linear hypergraphs. Computational methods have previously been used to verify the EFL conjecture for small hypergraphs. We verify and extend these results to larger values and discuss challenges and directions. Ours is the first computational approach to the EFL conjecture that allows producing independently verifiable, DRAT proofs.

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IJCAI Conference 2023 Conference Paper

Co-Certificate Learning with SAT Modulo Symmetries

Markus Kirchweger
Tomáš Peitl
Stefan Szeider

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.

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SAT Conference 2023 Conference Paper

IPASIR-UP: User Propagators for CDCL

Katalin Fazekas
Aina Niemetz
Mathias Preiner
Markus Kirchweger
Stefan Szeider
Armin Biere

Modern SAT solvers are frequently embedded as sub-reasoning engines into more complex tools for addressing problems beyond the Boolean satisfiability problem. Examples include solvers for Satisfiability Modulo Theories (SMT), combinatorial optimization, model enumeration and counting. In such use cases, the SAT solver is often able to provide relevant information beyond the satisfiability answer. Further, domain knowledge of the embedding system (e. g. , symmetry properties or theory axioms) can be beneficial for the CDCL search, but cannot be efficiently represented in clausal form. In this paper, we propose a general interface to inspect and influence the internal behaviour of CDCL SAT solvers. Our goal is to capture the most essential functionalities that are sufficient to simplify and improve use cases that require a more fine-grained interaction with the SAT solver than provided via the standard IPASIR interface. For our experiments, we extend CaDiCaL with our interface and evaluate it 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.

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SAT Conference 2023 Conference Paper

SAT-Based Generation of Planar Graphs

Markus Kirchweger
Manfred Scheucher
Stefan Szeider

To test a graph’s planarity in SAT-based graph generation we develop SAT encodings with dynamic symmetry breaking as facilitated in the SAT modulo Symmetry (SMS) framework. We implement and compare encodings based on three planarity criteria. In particular, we consider two eager encodings utilizing order-based and universal-set-based planarity criteria, and a lazy encoding based on Kuratowski’s theorem. The performance and scalability of these encodings are compared on two prominent problems from combinatorics: the computation of planar Turán numbers and the Earth-Moon problem. We further showcase the power of SMS equipped with a planarity encoding by verifying and extending several integer sequences from the Online Encyclopedia of Integer Sequences (OEIS) related to planar graph enumeration. Furthermore, we extend the SMS framework to directed graphs which might be of independent interest.

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SAT Conference 2022 Conference Paper

A SAT Attack on Rota's Basis Conjecture

Markus Kirchweger
Manfred Scheucher
Stefan Szeider

The SAT modulo Symmetries (SMS) is a recently introduced framework for dynamic symmetry breaking in SAT instances. It combines a CDCL SAT solver with an external lexicographic minimality checking algorithm. We extend SMS from graphs to matroids and use it to progress on Rota’s Basis Conjecture (1989), which states that one can always decompose a collection of r disjoint bases of a rank r matroid into r disjoint rainbow bases. Through SMS, we establish that the conjecture holds for all matroids of rank 4 and certain special cases of matroids of rank 5. Furthermore, we extend SMS with the facility to produce DRAT proofs. External tools can then be used to verify the validity of additional axioms produced by the lexicographic minimality check. As a byproduct, we have utilized our framework to enumerate matroids modulo isomorphism and to support the investigation of various other problems on matroids.

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