arXiv subjects

Ruben F. A. Verhaegh

Publications and source records attributed to Ruben F. A. Verhaegh.

4 recordsLinked to original sources

arXiv2026

Search-space Reduction for Boolean MinCSPs via Essential Constraints

For a fixed set $\mathcal{F}$ of Boolean constraint types, a MinCSP$(\mathcal{F})$-instance consists of a formula $F$ that applies $m$ constraints from $\mathcal{F}$ to a set of $n$ Boolean variables. The goal is to remove a minimum subset of constraint applications from $F$ to make the remaining formula satisfiable. Previous work characterized how the choice of $\mathcal{F}$ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula $F$ as $c$-essential if it is contained in all $c$-approximate solutions to $F$. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets $\mathcal{F}$ for which $c_\mathcal{F}$-essential constraint applications can be detected efficiently for some $c_{\mathcal{F}} \in \mathcal{O}(1)$, from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set $\mathcal{F}$ of bijunctive constraints, there is a polynomial-time algorithm that detects $\mathcal{O}(1)$-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP$(\mathcal{F})$-problem is intractable under the Unique Games Conjecture.

cs.CC↗

arXiv2025

An ETH-Tight FPT Algorithm for Rejection-Proof Set Packing with Applications to Kidney Exchange

We study the parameterized complexity of a recently introduced multi-agent variant of the Kidney Exchange problem. Given a directed graph $G$ and integers $d$ and $k$, the standard problem asks whether $G$ contains a packing of vertex-disjoint cycles, each of length $\leq d$, covering at least $k$ vertices in total. In the multi-agent setting we consider, the vertex set is partitioned over several agents who reject a cycle packing as solution if it can be modified into an alternative packing that covers more of their own vertices. A cycle packing is called rejection-proof if no agent rejects it and the problem asks whether such a packing exists that covers at least $k$ vertices. We exploit the sunflower lemma on a set packing formulation of the problem to give a kernel for this $Σ_2^P$-complete problem that is polynomial in $k$ for all constant values of $d$. We also provide a $2^{\mathcal{O}(k \log k)} + n^{\mathcal{O}(1)}$ algorithm based on it and show that this FPT algorithm is asymptotically optimal under the ETH. Further, we generalize the problem by including an additional positive integer $c$ in the input that naturally captures how much agents can modify a given cycle packing to reject it. For every constant $c$, the resulting problem simplifies from being $Σ_2^P$-complete to NP-complete. The super-exponential lower bound already holds for $c=2$, though. We present an ad-hoc single-exponential algorithm for $c = 1$. These results reveal an interesting discrepancy between the classical and parameterized complexity of the problem and give a good view of what makes it hard.

cs.DS↗

arXiv2024

Preprocessing to Reduce the Search Space for Odd Cycle Transversal

The NP-hard Odd Cycle Transversal problem asks for a minimum vertex set whose removal from an undirected input graph $G$ breaks all odd cycles, and thereby yields a bipartite graph. The problem is well-known to be fixed-parameter tractable when parameterized by the size $k$ of the desired solution. It also admits a randomized kernelization of polynomial size, using the celebrated matroid toolkit by Kratsch and Wahlström. The kernelization guarantees a reduction in the total $\textit{size}$ of an input graph, but does not guarantee any decrease in the size of the solution to be sought; the latter governs the size of the search space for FPT algorithms parameterized by $k$. We investigate under which conditions an efficient algorithm can detect one or more vertices that belong to an optimal solution to Odd Cycle Transversal. By drawing inspiration from the popular $\textit{crown reduction}$ rule for Vertex Cover, and the notion of $\textit{antler decompositions}$ that was recently proposed for Feedback Vertex Set, we introduce a graph decomposition called $\textit{tight odd cycle cut}$ that can be used to certify that a vertex set is part of an optimal odd cycle transversal. While it is NP-hard to compute such a graph decomposition, we develop parameterized algorithms to find a set of at least $k$ vertices that belong to an optimal odd cycle transversal when the input contains a tight odd cycle cut certifying the membership of $k$ vertices in an optimal solution. The resulting algorithm formalizes when the search space for the solution-size parameterization of Odd Cycle Transversal can be reduced by preprocessing. To obtain our results, we develop a graph reduction step that can be used to simplify the graph to the point that the odd cycle cut can be detected via color coding.

cs.DS↗

arXiv2024

Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion

For an optimization problem $Π$ on graphs whose solutions are vertex sets, a vertex $v$ is called $c$-essential for $Π$ if all solutions of size at most $c \cdot OPT$ contain $v$. Recent work showed that polynomial-time algorithms to detect $c$-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size $k$ of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for $3$-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected $n$-vertex graph $G$ in time $2^{O(\ell^3)} \cdot n^{O(1)}$, where $\ell$ is the number of vertices in an optimal solution that are not $3$-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of $c$, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that ($2-\varepsilon$)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects $2$-essential vertices is best-possible.

cs.DS↗