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.

Given a graph G and two independent sets of same size, the Independent Set Reconfiguration Problem under token sliding asks whether one can, in a step by step manner, transform the first independent set into the second one. In each step we must preserve the condition of independence. Further, referring to solution vertices as tokens, we are only permitted to slide a token along an edge. Until the recent work of Ito et al. [Ito et al. MFCS 2022] this problem was only considered on undirected graphs. In this work, we study reconfiguration under token sliding focusing on DAGs. We present a complete dichotomy of intractability in regard to the depth of the DAG, by proving that this problem is NP-complete for DAGs of depth 3 and W[1]-hard for depth 4 when parameterized by the number of tokens k, and that these bounds are tight. Further, we prove that it is fixed parameter tractable on DAGs parameterized by the combination of treewidth and k. We show that this result applies to undirected graphs, when the number of times a token can visit a vertex is restricted.

Abstract We study reduction rules for Directed Feedback Vertex Set (DFVS) on directed graphs without long cycles. A DFVS instance without cycles longer than d naturally corresponds to an instance of d -Hitting Set, however, enumerating all cycles in an n-vertex graph and then kernelizing the resulting d -Hitting Set instance can be too costly, as already enumerating all cycles can take time $$\Omega (n^d)$$ Ω ( n d ) . To the best of our knowledge, the kernelization of DFVS on graphs without long cycles has not been studied in the literature, except for very restricted cases, e.g., for tournaments, in which all induced cycles are of length three. We show that the natural reduction rule to delete all vertices and edges that do not lie on induced cycles cannot be implemented efficiently, that is, it is W[1]-hard (with respect to parameter d) to decide if a vertex or edge lies on an induced cycle of length at most d even on graphs that become acyclic after the deletion of a single vertex or edge. Based on different reduction rules we then show how to compute a kernel with at most $$2^dk^d$$ 2 d k d vertices and at most $$d^{3d}k^d$$ d 3 d k d induced cycles of length at most d (which however, cannot be enumerated efficiently), where k is the size of a minimum directed feedback vertex set. We then study classes of graphs whose underlying undirected graphs have bounded expansion or are nowhere dense. These are very general classes of sparse graphs, containing e.g. classes excluding a minor or a topological minor. We prove that for every class $$\mathscr {C} $$ C with bounded expansion there is a function $$f_\mathscr {C} (d)$$ f C ( d ) such that for graphs $$G\in \mathscr {C} $$ G ∈ C without induced cycles of length greater than d we can compute a kernel with $$f_\mathscr {C} (d)\cdot k$$ f C ( d ) · k vertices in time $$f_\mathscr {C} (d)\cdot n^{\mathcal {O}(1)}$$ f C ( d ) · n O ( 1 ) . For every nowhere dense class $$\mathscr {C} $$ C there is a function $$f_\mathscr {C} (d,\varepsilon )$$ f C ( d , ε ) such that for graphs $$G\in \mathscr {C} $$ G ∈ C without induced cycles of length greater than d we can compute a kernel with $$f_\mathscr {C} (d,\varepsilon )\cdot k^{1+\varepsilon }$$ f C ( d , ε ) · k

We present an exact solver for the DFVS, submitted for the exact track of the Parameterized Algorithms and Computational Experiments challenge (PACE) in 2022. The solver heavily relies on data reduction (known from the literature and new reduction rules). The instances are then further processed by integer linear programming approaches. We implemented the algorithm in the scope of a student project at the University of Bremen.