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Ciaran McCreesh

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15 papers

2 author rows

AAAI Conference 2026 Conference Paper

Faster Certified Symmetry Breaking Using Orders with Auxiliary Variables

Markus Anders
Bart Bogaerts
Benjamin Bogø
Arthur Gontier
Wietze Koops
Ciaran McCreesh
Magnus O. Myreen
Jakob Nordström

Symmetry breaking is a crucial technique in modern combinatorial solving, but it is difficult to be sure it is implemented correctly. The most successful approach to deal with bugs is to make solvers certifying, so that they output not just a solution, but also a mathematical proof of correctness in a standard format, which can then be checked by a formally verified checker. This requires justifying symmetry reasoning within the proof, but developing efficient methods for this has remained a long-standing open challenge. A fully general approach was recently proposed, but it relies on encoding lexicographic orders with big integers, which quickly becomes infeasible for large symmetries. In this work, we develop a method for instead encoding orders with auxiliary variables. We show that this leads to orders-of-magnitude speed-ups in both theory and practice by running experiments on proof logging and checking for SAT symmetry breaking using the state-of-the-art satsuma symmetry breaker and the VeriPB proof checking toolchain.

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AAAI Conference 2025 Conference Paper

Certifying Bounds Propagation for Integer Multiplication Constraints

Matthew J. McIlree
Ciaran McCreesh

A constraint programming (CP) solver that implements proof logging will output a machine-checkable certificate of correctness alongside any result it obtains. This is useful for trusting claims of unsatisfiability or optimality, as well as for debugging and auditing solver implementations. Proofs can be constructed by having the solver log justifications for each inference it makes, and previous work has shown that many standard CP reasoning techniques can be efficiently justified using a pseudo-Boolean (PB) proof format. This paper extends PB justifications to propagators enforcing bounds consistency on multiplication and division constraints. We show that even though the proof system and checker operate only on linear inequalities over 0-1 variables, non-linear reasoning over bounded domains can be efficiently expressed as a sequence of PB proof steps. Additionally, we demonstrate that bespoke proof logging for bounds-consistency algorithms offers a clear advantage over constructing justifications by brute force.

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AAAI Conference 2024 Conference Paper

End-to-End Verification for Subgraph Solving

Stephan Gocht
Ciaran McCreesh
Magnus O. Myreen
Jakob Nordström
Andy Oertel
Yong Kiam Tan

Modern subgraph-finding algorithm implementations consist of thousands of lines of highly optimized code, and this complexity raises questions about their trustworthiness. Recently, some state-of-the-art subgraph solvers have been enhanced to output machine-verifiable proofs that their results are correct. While this significantly improves reliability, it is not a fully satisfactory solution, since end-users have to trust both the proof checking algorithms and the translation of the high-level graph problem into a low-level 0-1 integer linear program (ILP) used for the proofs. In this work, we present the first formally verified toolchain capable of full end-to-end verification for subgraph solving, which closes both of these trust gaps. We have built encoder frontends for various graph problems together with a 0-1 ILP (a.k.a. pseudo-Boolean) proof checker, all implemented and formally verified in the CakeML ecosystem. This toolchain is flexible and extensible, and we use it to build verified proof checkers for both decision and optimization graph problems, namely, subgraph isomorphism, maximum clique, and maximum common (connected) induced subgraph. Our experimental evaluation shows that end-to-end formal verification is now feasible for a wide range of hard graph problems.

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IJCAI Conference 2024 Conference Paper

Proof Logging for Smart Extensional Constraints (Extended Abstract)

Matthew J. McIlree
Ciaran McCreesh

Proof logging provides an auditable way of guaranteeing that a solver has produced a correct answer using sound reasoning. This is standard practice for Boolean satisfiability solving, but for constraint programming, a challenge is that every propagator must be able to justify all inferences it performs. Here we demonstrate how to support proof logging for a wide range of previously uncertified global constraints. We do this by showing how to justify every inference that could be performed by the propagation algorithms for two families of generalised extensional constraint: "Smart Table" and "Regular Language Membership".

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JAIR Journal 2023 Journal Article

Certified Dominance and Symmetry Breaking for Combinatorial Optimisation

Bart Bogaerts
Stephan Gocht
Ciaran McCreesh
Jakob Nordström

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

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AAAI Conference 2022 Conference Paper

Certified Symmetry and Dominance Breaking for Combinatorial Optimisation

Bart Bogaerts
Stephan Gocht
Ciaran McCreesh
Jakob Nordström

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking are easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

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IJCAI Conference 2021 Conference Paper

Solving Graph Homomorphism and Subgraph Isomorphism Problems Faster Through Clique Neighbourhood Constraints

Sonja Kraiczy
Ciaran McCreesh

Graph homomorphism problems involve finding adjacency-preserving mappings between two given graphs. Although theoretically hard, these problems can often be solved in practice using constraint programming algorithms. We show how techniques from the state-of-the-art in subgraph isomorphism solving can be applied to broader graph homomorphism problems, and introduce a new form of filtering based upon clique-finding. We demonstrate empirically that this filtering is effective for the locally injective graph homomorphism and subgraph isomorphism problems, and gives the first practical constraint programming approach to finding general graph homomorphisms.

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AAAI Conference 2020 Conference Paper

Justifying All Differences Using Pseudo-Boolean Reasoning

Jan Elffers
Stephan Gocht
Ciaran McCreesh
Jakob Nordström

Constraint programming solvers support rich global constraints and propagators, which make them both powerful and hard to debug. In the Boolean satisﬁability community, prooflogging is the standard solution for generating trustworthy outputs, and this has become key to the social acceptability of computer-generated proofs. However, reusing this technology for constraint programming requires either much weaker propagation, or an impractical blowup in proof length. This paper demonstrates that simple, clean, and efﬁcient proof logging is still possible for the all-different constraint, through pseudo-Boolean reasoning. We explain how such proofs can be expressed and veriﬁed mechanistically, describe an implementation, and discuss the broader implications for proof logging in constraint programming.

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IJCAI Conference 2020 Conference Paper

Subgraph Isomorphism Meets Cutting Planes: Solving With Certified Solutions

Stephan Gocht
Ciaran McCreesh
Jakob Nordström

Modern subgraph isomorphism solvers carry out sophisticated reasoning using graph invariants such as degree sequences and path counts. We show that all of this reasoning can be justified compactly using the cutting planes proofs studied in complexity theory. This allows us to extend a state of the art subgraph isomorphism enumeration solver with proof logging support, so that the solutions it outputs may be audited and verified for correctness and completeness by a simple third party tool which knows nothing about graph theory.

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JAIR Journal 2018 Journal Article

When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases

Ciaran McCreesh
Patrick Prosser
Christine Solnon
James Trimble

The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate "really hard" random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable / unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrainedness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched "filter / verify" indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm.

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IJCAI Conference 2017 Conference Paper

A Partitioning Algorithm for Maximum Common Subgraph Problems

Ciaran McCreesh
Patrick Prosser
James Trimble

We introduce a new branch and bound algorithm for the maximum common subgraph and maximum common connected subgraph problems which is based around vertex labelling and partitioning. Our method in some ways resembles a traditional constraint programming approach, but uses a novel compact domain store and supporting inference algorithms which dramatically reduce the memory and computation requirements during search, and allow better dual viewpoint ordering heuristics to be calculated cheaply. Experiments show a speedup of more than an order of magnitude over the state of the art, and demonstrate that we can operate on much larger graphs without running out of memory.

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AAAI Conference 2017 Conference Paper

Between Subgraph Isomorphism and Maximum Common Subgraph

Ruth Hoffmann
Ciaran McCreesh
Craig Reilly

When a small pattern graph does not occur inside a larger target graph, we can ask how to ﬁnd “as much of the pattern as possible” inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph problem.

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SoCS Conference 2016 Conference Paper

Finding Maximum k-Cliques Faster Using Lazy Global Domination

Ciaran McCreesh
Patrick Prosser

A clique in a graph is a set of vertices, each of which is adjacent to every other vertex in this set. A k-clique relaxes this requirement, requiring vertices to be within a distance k of each other, rather than directly adjacent. In theory, a maximum clique algorithm can easily be adapted to solve the maximum k-clique problem, although large sparse k-clique graphs reduce to large dense clique graphs, which can be computationally challenging. We adapt a state of the art maximum clique algorithm to show that this reduction is in fact useful in practice, and introduce a lazy global domination rule which sometimes vastly reduces the search space. We include experimental results for a range of real-world and benchmark graphs, and a detailed look at random graphs. We also use thread-parallel search to solve some harder instances.

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IJCAI Conference 2016 Conference Paper

Heuristics and Really Hard Instances for Subgraph Isomorphism Problems

Ciaran McCreesh
Patrick Prosser
James Trimble

We show how to generate "really hard'" random instances for subgraph isomorphism problems. For the non-induced variant, we predict and observe a phase transition between satisfiable and unsatisfiable instances, with a corresponding complexity peak seen in three different solvers. For the induced variant, much richer behaviour is observed, and constrainedness gives a better measure of difficulty than does proximity to a phase transition. We also discuss variable and value ordering heuristics, and their relationship to the expected number of solutions.

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IJCAI Conference 2016 Conference Paper

Solving Hard Subgraph Problems in Parallel

Ciaran McCreesh

We look at problems involving finding subgraphs in larger graphs, such as the maximum clique problem, the subgraph isomorphism problem, and the maximum common subgraph problem. We investigate variable and value ordering heuristics, different inference strategies, intelligent backtracking search (backjumping), and bit- and thread-parallelism to exploit modern hardware.

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