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37th Annual Conference on Learning Theory, COLT 2024

作者： Chalopin, Jérémie Chepoi, Victor Mc Inerney, Fionn Ratel, Sébastien Aix-Marseille Université Université de Toulon CNRS LIS Marseille France Algorithms and Complexity Group TU Wien Vienna Austria

Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it to be the most efficient machine teaching model satisfying the benchmark for collusion-avoidance set by Goldman and Mathias. A teaching map T for a concept class C assigns a (teaching) set T(C) of examples to each concept C ∈ C. A teaching map is non-clashing if no pair of concepts are consistent with the union of their teaching sets. The size of a non-clashing teaching map (NCTM) T is the maximum size of a teaching set T(C), C ∈ C. The non-clashing teaching dimension NCTD(C) of C is the minimum size of an NCTM for C. NCTM+ and NCTD+(C) are defined analogously, except the teacher may only use positive examples. We study NCTMs and NCTM+s for the concept class B(G) consisting of all balls of a graph G. We show that the associated decision problem B-NCTD+ for NCTD+ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, B-NCTD+ does not admit an algorithm running in time 22o(vc) ·nO(1), nor a kernelization algorithm outputting a kernel with 2o(vc) vertices, where vc is the vertex cover number of G. We complement these lower bounds with matching upper bounds. These are extremely rare results: it is only the second problem in NP to admit such a tight double-exponential lower bound parameterized by vc, and only one of very few problems to admit such an ETH-based conditional lower bound on the number of vertices in a kernel. For trees, interval graphs, cycles, and trees of cycles, we derive NCTM+s or NCTMs for B(G) of size proportional to its VC-dimension. For Gromov-hyperbolic graphs, we design an approximate NCTM+ for B(G) of size 2, in which only pairs of balls with Hausdorff distance larger than some constant must satisfy the non-clashing condition. © 2024 J. Chalopin, V. Chepoi, F. Mc Inerney & S. Ratel.

关键词： Parameterization

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A Simple Partially Embedded Planarity Test Based on Vertex-Addition 8

A Simple Partially Embedded Planarity Test Based on Vertex-A...

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8th SIAM Symposium on Simplicity of algorithms, SOSA 2025

作者： Fink, Simon D. Rutter, Ignaz Sandhya, T.P. Algorithms and Complexity Group Technische Universität Wien Austria Faculty of Informatics and Mathematics University of Passau Germany Stockholm University Department of Mathematics Sweden

ISBN: (纸本)9781611978315

In the Partially Embedded Planarity problem, we are given a graph G together with a topological drawing of a subgraph H of G. The task is to decide whether the drawing can be extended to a drawing of the whole graph such that no two edges cross. Angelini et al. gave a linear-time algorithm for solving this problem in 2010 [1, 2]. While their paper constitutes a significant result, the algorithm described therein is highly complex: it uses several layers of decompositions according to connectivity of both G and H, its description spans more than 30 pages, and can hardly be considered implementable. We give an independent linear-time algorithm that works along the well-known vertex-addition planarity test by Booth and Lueker [5, 6]. We modify the PC-tree as underlying data structure used for representing all planar drawing possibilities in a natural way to also respect the restrictions given by the prescribed drawing of the subgraph H. The testing algorithm and its proof of correctness only require small adaptations from the comparatively much simpler generic planarity test, of which several implementations exist. If the test succeeds, an embedding can be constructed using the same approaches that are used for the generic planarity test. Copyright © 2025 by SIAM.

关键词： Trees (mathematics)

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Deterministic Constrained Multilinear Detection 48

Deterministic Constrained Multilinear Detection

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48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023

作者： Brand, Cornelius Korchemna, Viktoriia Skotnica, Michael Algorithms and Complexity Group TU Wien Austria Department of Applied Mathematics Charles University Prague Czech Republic

ISBN: (纸本)9783959772921

We extend the algebraic techniques of Brand and Pratt (ICALP’21) for deterministic detection of k-multilinear monomials in a given polynomial with non-negative coefficients to the more general situation of detecting colored k-multilinear monomials that satisfy additional constraints on the multiplicities of the colors appearing in them. Our techniques can be viewed as a characteristic-zero generalization of the algebraic tools developed by Guillemot and Sikora (MFCS’10) and Björklund, Kaski and Kowalik (STACS’13) As applications, we recover the state-of-the-art deterministic algorithms for the Graph Motif problem due to Pinter, Schachnai and Zehavi (MFCS’14), and give new deterministic algorithms for generalizations of certain questions on colored directed spanning trees or bipartite planar matchings running in deterministic time O∗(4k), studied originally by Gutin, Reidl, Wahlström and Zehavi (J. Comp. Sys. Sci. 95,’18). Finally, we give improved randomized algorithms for intersecting three and four matroids of rank k in characteristic zero, improving the record bounds of Brand and Pratt (ICALP’21) from O∗(64k) and O∗(256k), respectively, to O∗(4k). © Cornelius Brand, Viktoriia Korchemna, and Michael Skotnica

关键词： Trees (mathematics)

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Parameterized complexity of Efficient Sortation

arXiv

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arXiv 2024年

作者： Ganian, Robert Hoang, Hung P. Wietheger, Simon TU Vienna Algorithms and Complexity Group Austria

A crucial challenge arising in the design of large-scale logistical networks is to optimize parcel sortation for routing. We study this problem under the recent graph-theoretic formalization of Van Dyk, Klause, Koenemann and Megow (IPCO 2024). The problem asks-given an input digraph D (the fulfillment network) together with a set of commodities represented as source-sink tuples-for a minimum-outdegree subgraph H of the transitive closure of D that contains a source-sink route for each of the commodities. Given the underlying motivation, we study two variants of the problem which differ in whether the routes for the commodities are assumed to be given, or can be chosen arbitrarily. We perform a thorough parameterized analysis of the complexity of both problems. Our results concentrate on three fundamental parameterizations of the problem: 1. When attempting to parameterize by the target outdegree of H, we show that the problems are paraNP-hard even in highly restricted cases;2. When parameterizing by the number of commodities, we utilize Ramsey-type arguments and color-coding techniques to obtain parameterized algorithms for both problems;3. When parameterizing by the structure of D, we establish fixed-parameter tractability for both problems w.r.t. treewidth, maximum degree and the maximum routing length. We combine this with lower bounds which show that omitting any of the three parameters results in paraNPhardness. © 2024, CC BY.

关键词： Parameterization

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Extracting Problem Structure with LLMs for Optimized SAT Local Search

arXiv

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arXiv 2025年

作者： Schidler, André Szeider, Stefan Algorithms and Complexity Group TU Wien Vienna Austria

Local search preprocessing makes Conflict-Driven Clause Learning (CDCL) solvers faster by providing high-quality starting points and modern SAT solvers have incorporated this technique into their preprocessing steps. However, these tools rely on basic strategies that miss the structural patterns in problems. We present a method that applies Large Language Models (LLMs) to analyze Python-based encoding code. This reveals hidden structural patterns in how problems convert into SAT. Our method automatically generates specialized local search algorithms that find these patterns and use them to create strong initial assignments. This works for any problem instance from the same encoding type. Our tests show encouraging results, achieving faster solving times compared to baseline preprocessing systems. Copyright © 2025, The Authors. All rights reserved.

关键词： Encoding (symbols)

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Smart Cubing for Graph Search: A Comparative Study

arXiv

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arXiv 2025年

作者： Kirchweger, Markus Xia, Hai Peitl, Tomáš Szeider, Stefan Algorithms and Complexity Group TU Wien Vienna Austria

Parallel solving via cube-and-conquer is a key method for scaling SAT solvers to hard instances. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers extended with propagators presents unique challenges, as these propagators learn constraints dynamically during the search. We study this problem using SAT Modulo Symmetries (SMS) as our primary test case, where a symmetry-breaking propagator reduces the search space by learning constraints that eliminate isomorphic graphs. Through extensive experimentation comprising over 10,000 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 and LLM-generated design suggestions. The comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving, with our best method achieving speedups of 2-3x from improved cubing and parameter tuning, providing an additional 1.5-2x improvement on harder instances. Copyright © 2025, The Authors. All rights reserved.

关键词： Geometry

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The Power of Collaboration: Learning Large Bayesian Networks at Scale

The Power of Collaboration: Learning Large Bayesian Networks...

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International Conference on Tools for Artificial Intelligence (ICTAI)

作者： Vaidyanathan Peruvemba Ramaswamy Stefan Szeider Hai Xia Algorithms and Complexity Group TU Wien Vienna Austria

ISBN: (数字)9798331527235

ISBN: (纸本)9798331527242

We present a novel approach to learn the structure of large Bayesian Networks (BNs) of bounded treewidth. Our approach combines the complementary strengths of heuristic and MaxSAT-based methods. Both methods operate in parallel and cooperate in a mutually advantageous way to improve the quality of the learned BN structures. Our work utilizes an approach proposed by Peruvemba Ramaswamy and Szeider (AAAI'21, NeurIPS'21, UAI'22), which continually applies a MaxSAT-based algorithm to a BN obtained from heuristic search. We create a continuous and dynamic refinement process by allowing the heuristic and the MaxSAT-based technique to cooperate and repeatedly share their work. Our rigorous experiments show that the collaboration between the two methods is a powerful strategy for increasing the score of the learned BN structure. Moreover, the rate at which the score increases is significantly higher than that of the individual methods by themselves. Our results provide a strong argument for combining complementary approaches to learn treewidth-bounded BNs in a collaborative way.

关键词： Image color analysis Heuristic algorithms Collaboration Real-time systems Bayes methods Relays Artificial intelligence Tuning Optimization Bars

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The Parameterized complexity Landscape of the Unsplittable Flow Problem

arXiv

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arXiv 2024年

作者： Ganian, Robert Rocton, Mathis Unterberger, Daniel Algorithms and Complexity Group TU Wien Vienna Austria

We study the well-established problem of finding an optimal routing of unsplittable flows in a graph. While by now there is an extensive body of work targeting the problem on graph classes such as paths and trees, we aim at using the parameterized paradigm to identify its boundaries of tractability on general graphs. We develop novel algorithms and lower bounds which result in a full classification of the parameterized complexity of the problem with respect to natural structural parameterizations for the problem—notably maximum capacity, treewidth, maximum degree, and maximum flow length. In particular, we obtain a fixed-parameter algorithm for the problem when parameterized by all four of these parameters, establish XP-tractability as well as W[1]hardness with respect to the former three and latter three parameters, and all remaining cases remain paraNP-hard. © 2024, CC BY.

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Compilation and Fast Model Counting beyond CNF

arXiv

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arXiv 2025年

作者： de Colnet, Alexis Szeider, Stefan Zhang, Tianwei Algorithms and Complexity Group TU Wien Vienna Austria

Circuits in deterministic decomposable negation normal form (d-DNNF) are representations of Boolean functions that enable linear-time model counting. This paper strengthens our theoretical knowledge of what classes of functions can be efficiently transformed, or compiled, into d-DNNF. Our main contribution is the fixed-parameter tractable (FPT) compilation of conjunctions of specific constraints parameterized by incidence treewidth. This subsumes the known result for CNF. The constraints in question are all functions representable by constant-width ordered binary decision diagrams (OBDDs) for all variable orderings. For instance, this includes parity constraints and cardinality constraints with constant threshold. The running time of the FPT compilation is singly exponential in the incidence treewidth but hides large constants in the exponent. To balance that, we give a more efficient FPT algorithm for model counting that applies to a sub-family of the constraints and does not require compilation. © 2025, CC BY.

关键词： Boolean functions

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SAT-Based Generation of Planar Graphs 26

SAT-Based Generation of Planar Graphs

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26th International Conference on Theory and Applications of Satisfiability Testing, SAT 2023

作者： Markus Kirchweger Manfred Scheucher Stefan Szeider Algorithms and Complexity Group TU Wien Austria Institut för Mathematik Technische Universität Berlin Germany

ISBN: (纸本)9783959772860

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. © 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

关键词： Moon

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