This paper describes our hybrid MaxSAT and mixed integer programming approach for finding minimum hitting sets as submitted to the 2025 PACE challenge. We also discuss hitting set specific challenges, lower bounds, preprocessing and design choices. 2012 ACM Subject Classification Theory of computation -> Discrete optimization; Theory of computation -> Constraint and logic programming; Theory of computation -> Automated reasoning
We investigate the enumeration of dense subgraphs under two well-known relaxations of cliques: k-plexes and k-defective cliques. Our main contribution is a family of algorithms with improved worst-case and output-sensitive complexities, driven by a decomposition technique based on graph degeneracy. We first propose a worst-case output-size near-optimal algorithm to enumerate all maximal k-plexes of size at least 2k - 1, achieving a total time complexity of O(n(dk)(3)2(d)Delta(k)), where d is the degeneracy and Delta the maximum degree of the input graph. We then refine this result to obtain a fixed-parameter tractable output-sensitive algorithm with complexity O(alpha f(k)p(d Delta)), where alpha is the number of solutions, f(k) is an arbitrary function of k, and p is a polynomial. We then extend this framework to the enumeration of k-defective cliques and also show a linear-time O(n) algorithm for the enumeration of 2-plexes for graphs with bounded degeneracy. To the best of our knowledge, these complexities are competitive with or better than the current state of the art. 2012 ACM Subject Classification Theory of computation. Parameterized complexity and exact algorithms
We present and describe the solver OBLX for the Dominating Set problem on graphs. This solver was developed during the PACE challenge 2025 for the Exact track. It first applies several data reduction rules and performs a polynomial time reduction to Max Sat. The resulting Max Sat instance is in turn solved using the EvalMaxSat solver by Florent Avellaneda. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
This article briefly describes the most important algorithms and techniques used in HitS&DoSeS, a dominating set and hitting set solver submitted to the PACE 2025 contest (10th Parameterized Algorithms and Computational Experiments Challenge). Used approaches for the exact and heuristic tracks are described, for both the dominating set and the hitting set problems. 2012 ACM Subject Classification Mathematics of computing -> Graph algorithms
Dominating Set and Hitting Set are two well-known NP-hard problems on graphs and hypergraphs, respectively. For Dominating Set, we seek a subset S of vertices of minimum size, such that every vertex has a neighbor in S. For Hitting Set, we require that this minimum size subset S intersects each hyperedge. We present Bad Dominating Set Maker, our solver for both problems posed in the exact tracks of the 2025 PACE Challenge. It uses reduction rules, dynamic programming on tree decompositions, and external Vertex Cover and SAT solvers. 2012 ACM Subject Classification Theory of computation -> Graph algorithms analysis; Theory of computation -> Parameterized complexity and exact algorithms
In this paper, we describe the solver we submitted for both heuristic tracks of the PACE challenge 2025 on the dominating set problem and the hitting set problem. We solve both problems as unicost set covering problems. Our solver first performs reductions on the instance. Then greedy algorithms generate an initial solution that serves as starting point of the large neighborhood search and the local search which are executed afterwards. The solver ranked first in the dominating set heuristic track, and second in the hitting set heuristic track. 2012 ACM Subject Classification Mathematics of computing -> Combinatorial optimization
We present a unified heuristic solver for the PACE 2025 challenge, addressing both the dominating set and hitting set problems by reducing them to the unicost set covering problem. Our solver applies standard reduction rules, a multi-round frequency-based greedy initializer, and a local search guided by adaptive element weights. Additional techniques, such as component-level exact solving and swap restriction, further enhance performance. In the final official evaluation, our proposed solver achieved second place in the heuristic track for the dominating set problem of the PACE 2025 challenge, while securing first place in the heuristic track for the hitting set problem. 2012 ACM Subject Classification Mathematics of computing -> Combinatorial optimization; Computing methodologies -> Search methodologies
The 10th iteration of the of the Parameterized Algorithms and Computational Experiments challenge (PACE) 2025 was devoted to engineer algorithms solving the Dominating Set problem as well as the Hitting Set problem. In contrast to the last iterations, these problems are (under standard assumptions) not fixed-parameter tractable (fpt) in general. However, restricting the structure of the input (e.g. to planar graphs or degenerate graphs for Dominating Set, or to set systems with sets of bounded size for Hitting Set) renders these problems fpt. Following the spirit of the last iterations of the PACE challenge, there is an exact track and a heuristic track for each problem; each track coming with a benchmark set of 100 public instances and 100 private instances. Overall, the PACE 2025 had 71 participants from 25 teams, 13 countries, and 3 continents. In this report, we briefly describe the setup of the challenge, the selection of benchmark instances, as well as the ranking of the participating teams. We also briefly outline the approaches used in the submitted solvers. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms; Theory of computation -> Graph algorithms analysis
Many classical graph problems - such as Max Cut, Chromatic Number, Edge Dominating Set, and Hamiltonian Cycle - are polynomial-time solvable on cographs, fixed-parameter tractable (FPT) when parameterized by treewidth, but W[1]-hard when parameterized by clique-width. In contrast, Graph Isomorphism is FPT parameterized by treewidth, but for clique-width it is known to be in XP; whether it is FPT or W[1]-hard is open. This reveals a sharp tractability gap between treewidth and clique-width. In this work, we propose a new structural graph parameter, C-modular-treewidth, which lies between treewidth and clique-width. The parameter leverages modular decomposition and restricts modules to induce graphs from a fixed class C (e.g., cographs or edgeless graphs). By exploiting true and false twins a hallmark of cograph-like structure - our parameter allows the design of efficient algorithms for several hard problems beyond the reach of treewidth-based methods. In this work, we show that C-modular-treewidth enables efficient solutions under suitable choices of C, opening a new pathway in the parameterized complexity landscape between treewidth and clique-width. In particular we show that When parameterized by cograph-modular-treewidth, Isomorphism admits an FPT algorithm, whereas Chromatic Number remains W[1]-hard. When parameterized by independent-modular-treewidth, Hamiltonian Cycle and Edge Dominating Set remain W[1]-hard. 2012 ACM Subject Classification Mathematics of computing -> Graph algorithms
Recently, Bojikian and Kratsch [ICALP 2024] presented a novel approach to tackle connectivity problems parameterized by clique-width (cw), based on counting (modulo 2) the number of representations of partial solutions, while allowing for possibly multiple representations to exist for the same partial solution. Using this technique, they got a SETH-tight bound of O-* (3(cw)) for the Steiner Tree problem, which was left open by Hegerfeld and Kratsch [ESA 2023]. We use the same technique to solve the Connected Odd Cycle Transversal problem in time O-* (12(cw)). Moreover, we prove that our result is tight by providing a SETH-based lower bound excluding algorithms with running time O-* ((12 - )(cw)). This answers another question of Hegerfeld and Kratsch [ESA 2023]. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
In this paper, we give new and faster deterministic algorithms to count the number of k-paths and trees in host graphs of bounded treewidth. Our algorithms use time that is single-exponential in the treewidth, and employ the determinant method from [4]. Modifications of the algorithms count in single-exponential time the number of k-paths between specified end-points, the number of k-cycles, and the number of trees with k vertices that are a subgraph of the host graph. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
Integer Programming (IP) is a fundamental but computationally hard problem. Still, certain efficiently solvable subclasses have been identified over time, most notably totally unimodular IPs in the 1950s, and fixed-dimension IPs in the 1980s. Starting around the year 2000, a stream of research has identified block-structured IPs as yet another tractable subclass. In this paper, we give a brief and incomplete review of this history, with a focus on several of the author's contributions. 2012 ACM Subject Classification Theory of computation -> Convex optimization
We consider the Maximum 2-Club problem where one is given as input an undirected graph G = ( V, E) and seeks a subset of vertices S of maximum size such that any pair of vertices in S is connected by a path of length at most 2 in the graph induced by S. This problem is a natural relaxation of the famous Maximum Clique problem where any pair of vertices must be connected by an edge. Maximum 2-Club has been well-studied and is known to be NP-complete even on split graphs. It can be solved exactly in O-*(1.62(n)) time, where n denotes the number of vertices of the input graph, while being polynomial-time solvable on several graph classes. Parameterized algorithms for structural parameters have also been considered, leading in particular to an algorithm with a double-exponential dependence in the parameter treewidth. Such an algorithm is actually the best one known for the larger parameter vertex cover size up to a constant in the exponent. We provide new results in both directions. We first prove that the double-exponential dependence for parameter vertex cover size is unavoidable under the Exponential Time Hypothesis (ETH). This answers a question left open by Hartung, Komusiewicz, Nichterlein and Suchy [14]. Our result also implies that the problem cannot be solved in time sub-exponential in n even for split graphs. We then provide an exact algorithm for the problem restricted to chordal graphs, running in O-*(1.1996(n)) time, by reducing Maximum 2-Club on this class to Maximum Independent Set on arbitrary graphs with the same number of vertices. The same reduction shows that we can enumerate all maximum (and inclusion-wise maximal) 2-clubs of a chordal graph in O*(3 (n/3)) = O *(1.4423(n)) time. We conclude by providing a construction of split graphs with Omega(3 (n/3) / poly(n)) maximum 2-clubs, for some polynomial poly showing that the bound for enumeration is essentially tight. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
Description of the solvers used by the Shadoks team in the PACE 2025 challenge. The challenge considers solvers for the minimum dominating set and hitting set problems. For the heuristic challenge, we respectively won third and fourth place for hitting set and dominating set. For the exact challenge, we won fifth place on both problems. 2012 ACM Subject Classification Theory of computation. Design and analysis of algorithms
This document contains a short description of our solver for the dominating set and hitting set problems that we submitted to the exact tracks of the PACE Challenge 2025. The solver is based on a straightforward MaxSAT formulation supplemented by hitting-set-based reduction rules. It utilizes a clique solver if the reduced instance is a (small) input for the vertex cover problem and tries to match certain lower bounds by expressing the reduced instance as a sat problem. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
The Boolean connectivity problem asks whether the set of satisfying assignments of a given Boolean formula forms a connected subgraph in the n-dimensional hypercube. This problem is known to be coNP-complete, even when restricted to k-Horn formulas for k >= 3, as shown by Makino, Tamaki, and Yamamoto. In this paper, we further investigate the complexity of the Boolean connectivity problem for k-Horn formulas, referred to as Conn k-Horn. We first present an exact exponential-time algorithm for Conn k-Horn without any structural restrictions. Our algorithm builds on the deterministic PPZ algorithm proposed by Paturi, Pudlak, and Zane. It runs in O-*(2((1-1/2k)n)) time, achieving an exponential improvement over the previously known algorithm for the Boolean connectivity problem of k-CNF formulas, shown by Makino, Tamaki, and Yamamoto. We then examine both algorithmic and hardness results for Conn 3-Horn under bounded variable occurrences. On the algorithmic side, we propose a polynomial-time algorithm for Conn 3-Horn when each clause contains exactly three literals and each variable appears at most three times. This result generalizes to Conn k-Horn under the same structural constraints, in which each clause contains exactly k literals and each variable appears at most k times. On the hardness side, we prove that Conn 3-Horn remains coNP-complete even when restricted to instances in which each variable appears exactly four times. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
This paper investigates the complexity of finding secluded paths in graphs. We focus on the Short Secluded Path problem and a natural new variant we introduce, Shortest Secluded Path. Formally, given an undirected graph G = ( V, E), two vertices s, t is an element of V, and two integers k, l, the Short Secluded Path problem asks whether there exists an s- t path of length at most k with at most l neighbors. This problem is known to be computationally hard: it is W[1]-hard when parameterized by the path length k or by cliquewidth, and para-NP-complete when parameterized by the number l of neighbors. The fixed-parameter tractability is known for k + l or treewidth. In this paper, we expand the parameterized complexity landscape by designing (1) an XP algorithm parameterized by cliquewidth and (2) fixed-parameter algorithms parameterized by neighborhood diversity and twin cover number, respectively. As a byproduct, our results also provide parameterized algorithms for the classic s- t k-Path problem. Furthermore, we introduce the Shortest Secluded Path problem, which seeks a shortest s- t path with the minimum number of neighbors. In contrast to the hardness of the original problem, we reveal that this variant is solvable in polynomial time on unweighted graphs. We complete this by showing that for edge-weighted graphs, the problem becomes W[1]-hard yet remains in XP when parameterized by the shortest path distance between s and t. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms
We investigate the parameterised complexity of the classic coverability problem for vector addition systems (VAS): given a finite set of vectors V subset of Z(d), an initial configuration s is an element of Nd, and a target configuration t is an element of Nd, decide whether starting from s, one can iteratively add vectors from V to ultimately arrive at a configuration that is larger than or equal to t on every coordinate, while not observing any negative value on any coordinate along the way. We consider two natural parameters for the problem: the dimension d and the size of V, defined as the total bitsize of its encoding. We present several results charting the complexity of those two parameterisations, among which the highlight is that coverability for VAS parameterised by the dimension and with all the numbers in the input encoded in unary is complete for the class XNL under PL-reductions. We also discuss open problems in the topic, most notably the question about fixed-parameter tractability for the parameterisation by the size of V. 2012 ACM Subject Classification Theory of computation -> Parameterized complexity and exact algorithms; Software and its engineering Model checking; Theory of computation -> Concurrency
We study the parameterized complexity of scheduling unit-time jobs on parallel, identical machines under generalized precedence constraints for minimization of the makespan and the sum of completion times ( P|gen-prec, p(j) = 1|y, y is an element of {C-max, Sigma(j) C-j}). In our setting, each job is equipped with a Boolean formula (precedence constraint) over the set of jobs. A schedule satisfies a job's precedence constraint if setting earlier jobs to true satisfies the formula. Our definition generalizes several common types of precedence constraints: classical and-constraints if every formula is a conjunction, or-constraints if every formula is a disjunction, and and/or-constraints if every formula is in conjunctive normal form. We prove fixed-parameter tractability when parameterizing by the number of predecessors. For parameterization by the number of successors, however, the complexity depends on the structure of the precedence constraints. If every constraint is a conjunction or a disjunction, we prove the problem to be fixed-parameter tractable. For constraints in disjunctive normal form, we prove W[1]-hardness. We show that the and/or-constrained problem is NP-hard, even for a single successor. Moreover, we prove NP-hardness on two machines if every constraint is a conjunction or a disjunction. This result not only proves para-NP-hardness for parameterization by the number of machines but also complements the polynomial-time solvability on two machines if every constraint is a conjunction ([4]) or if every constraint is a disjunction ([11]). 2012 ACM Subject Classification Mathematics of computing -> Combinatorial optimization
