arXiv subjects

Marijn Heule

Publications and source records attributed to Marijn Heule.

12 recordsLinked to original sources

arXiv2025

On the Edge of Core (Non-)Emptiness: An Automated Reasoning Approach to Approval-Based Multi-Winner Voting

Core stability is a natural and well-studied notion for group fairness in multi-winner voting, where the task is to select a committee from a pool of candidates. We study the setting where voters either approve or disapprove of each candidate; here, it remains a major open problem whether a core-stable committee always exists. In this work, we develop an approach based on mixed-integer linear programming for deciding whether and when core-stable committees are guaranteed to exist. In contrast to SAT-based approaches popular in computational social choice, our method can produce proofs for a specific number of candidates independent of the number of voters. In addition to these computational gains, our program lends itself to a novel duality-based reformulation of the core stability problem, from which we obtain new existence results in special cases. Further, we use our framework to reveal previously unknown relationships between core stability and other desirable properties, such as notions of priceability.

cs.GT↗

arXiv2022

Tighter Bounds on Directed Ramsey Number R(7)

Tournaments are orientations of the complete graph, and the directed Ramsey number $R(k)$ is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size $k$, which we denote by $TT_k$. We include a computer-assisted proof of a conjecture by Sanchez-Flores that all $TT_6$-free tournaments on 24 and 25 vertices are subtournaments of $ST_{27}$, the unique largest TT_6-free tournament. We also classify all $TT_6$-free tournaments on 23 vertices. We use these results, combined with assistance from SAT technology, to obtain the following improved bounds: $34 \leq R(7) \leq 47$.

math.CO↗

arXiv2022

An Impossible Asylum

In 1982, Raymond Smullyan published an article, "The Asylum of Doctor Tarr and Professor Fether," that consists of a series of puzzles. These were later reprinted in the anthology, "The Lady or The Tiger? and Other Logic Puzzles." The last puzzle, which describes the asylum alluded to in the title, was designed to be especially difficult. With the help of automated reasoning, we show that the puzzle's hypotheses are, in fact, inconsistent, which is to say, no such asylum can possibly exist.

math.LO↗

arXiv2020

Avoiding Monochromatic Rectangles Using Shift Patterns

Ramsey Theory deals with avoiding certain patterns. When constructing an instance that avoids one pattern, it is observed that other patterns emerge. For example, repetition emerges when avoiding arithmetic progression (Van der Waerden numbers), while reflection emerges when avoiding monochromatic solutions of $a+b=c$ (Schur numbers). We exploit observed patterns when coloring a grid while avoiding monochromatic rectangles. Like many problems in Ramsey Theory, this problem has a rapidly growing search space that makes computer search difficult. Steinbach et al. obtained a solution of an 18 by 18 grid with 4 colors by enforcing a rotation symmetry. However, that symmetry is not suitable for 5 colors. In this article, we will encode this problem into propositional logic and enforce so-called internal symmetries, which preserves satisfiability, to guide SAT-solving. We first observe patterns with 2 and 3 colors, among which the "shift pattern" can be easily generalized and efficiently encoded. Using this pattern, we obtain a new solution of the 18 by 18 grid that is non-isomorphic to the known solution. We further analyze the pattern and obtain necessary conditions to further trim down the search space. We conclude with our attempts on finding a 5-coloring of a 26 by 26 grid, as well as further open problems on the shift pattern.

cs.DM↗

arXiv2019

Constructing Minimal Perfect Hash Functions Using SAT Technology

Minimal perfect hash functions (MPHFs) are used to provide efficient access to values of large dictionaries (sets of key-value pairs). Discovering new algorithms for building MPHFs is an area of active research, especially from the perspective of storage efficiency. The information-theoretic limit for MPHFs is 1/(ln 2) or roughly 1.44 bits per key. The current best practical algorithms range between 2 and 4 bits per key. In this article, we propose two SAT-based constructions of MPHFs. Our first construction yields MPHFs near the information-theoretic limit. For this construction, current state-of-the-art SAT solvers can handle instances where the dictionaries contain up to 40 elements, thereby outperforming the existing (brute-force) methods. Our second construction uses XOR-SAT filters to realize a practical approach with long-term storage of approximately 1.83 bits per key.

cs.LO↗

arXiv2019

The Resolution of Keller's Conjecture

We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.

math.CO↗

arXiv2019

Computing properties of stable configurations of thermodynamic binding networks

The promise of chemical computation lies in controlling systems incompatible with traditional electronic micro-controllers, with applications in synthetic biology and nano-scale manufacturing. Computation is typically embedded in kinetics---the specific time evolution of a chemical system. However, if the desired output is not thermodynamically stable, basic physical chemistry dictates that thermodynamic forces will drive the system toward error throughout the computation. The thermodynamic binding network (TBN) model was introduced to formally study how the thermodynamic equilibrium can be made consistent with the desired computation, and it idealizes tradeoffs between configurational entropy and binding. Here we prove the computational hardness of natural questions about TBNs and develop a practical algorithm for verifying the correctness of constructions by translating the problem into propositional logic and solving the resulting formula. The TBN model together with automated verification tools will help inform strategies for error reduction in molecular computing, including the extensively studied models of strand displacement cascades and algorithmic tile assembly.

cs.ET↗

arXiv2016

Efficient Certified RAT Verification

Clausal proofs have become a popular approach to validate the results of SAT solvers. However, validating clausal proofs in the most widely supported format (DRAT) is expensive even in highly optimized implementations. We present a new format, called LRAT, which extends the DRAT format with hints that facilitate a simple and fast validation algorithm. Checking validity of LRAT proofs can be implemented using trusted systems such as the languages supported by theorem provers. We demonstrate this by implementing two certified LRAT checkers, one in Coq and one in ACL2.

cs.LO↗

arXiv2014

Towards Ultra Rapid Restarts

We observe a trend regarding restart strategies used in SAT solvers. A few years ago, most state-of-the-art solvers restarted on average after a few thousands of backtracks. Currently, restarting after a dozen backtracks results in much better performance. The main reason for this trend is that heuristics and data structures have become more restart-friendly. We expect further continuation of this trend, so future SAT solvers will restart even more rapidly. Additionally, we present experimental results to support our observations.

cs.AI↗

arXiv2014

Symbiosis of Search and Heuristics for Random 3-SAT

When combined properly, search techniques can reveal the full potential of sophisticated branching heuristics. We demonstrate this observation on the well-known class of random 3-SAT formulae. First, a new branching heuristic is presented, which generalizes existing work on this class. Much smaller search trees can be constructed by using this heuristic. Second, we introduce a variant of discrepancy search, called ALDS. Theoretical and practical evidence support that ALDS traverses the search tree in a near-optimal order when combined with the new heuristic. Both techniques, search and heuristic, have been implemented in the look-ahead solver march. The SAT 2009 competition results show that march is by far the strongest complete solver on random k-SAT formulae.

cs.DS↗

arXiv2010

Covered Clause Elimination

Generalizing the novel clause elimination procedures developed in [M. Heule, M. Järvisalo, and A. Biere. Clause elimination procedures for CNF formulas. In Proc. LPAR-17, volume 6397 of LNCS, pages 357-371. Springer, 2010.], we introduce explicit (CCE), hidden (HCCE), and asymmetric (ACCE) variants of a procedure that eliminates covered clauses from CNF formulas. We show that these procedures are more effective in reducing CNF formulas than the respective variants of blocked clause elimination, and may hence be interesting as new preprocessing/simplification techniques for SAT solving.

cs.LO↗

arXiv2010

Symmetry within Solutions

We define the concept of an internal symmetry. This is a symmety within a solution of a constraint satisfaction problem. We compare this to solution symmetry, which is a mapping between different solutions of the same problem. We argue that we may be able to exploit both types of symmetry when finding solutions. We illustrate the potential of exploiting internal symmetries on two benchmark domains: Van der Waerden numbers and graceful graphs. By identifying internal symmetries we are able to extend the state of the art in both cases.

cs.AI↗