arXiv subjects

Jaroslav Garvardt

Publications and source records attributed to Jaroslav Garvardt.

3 recordsLinked to original sources

arXiv2026

Clustering with Locally Bounded Ignorance

In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $\omega: {V \choose 2}\to Z$ and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most $k$ is FPT with respect to $k$ but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by $k+d$, where $d$ is the degeneracy of the fuzzy edge graph, and when parameterized by $k+c$, where $c$ is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.

cs.DS↗

arXiv2024

Parameterized Local Search for Max $c$-Cut

In the NP-hard Max $c$-Cut problem, one is given an undirected edge-weighted graph $G$ and aims to color the vertices of $G$ with $c$ colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with $c=2$ is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS Max $c$-Cut where we are also given a vertex coloring and an integer $k$ and the task is to find a better coloring that changes the color of at most $k$ vertices, if such a coloring exists; otherwise, the given coloring is $k$-optimal. We show that, for all $c\ge 2$, LS Max $c$-Cut presumably cannot be solved in $f(k)\cdot n^{\mathcal{O}(1)}$ time even on bipartite graphs. We then present an algorithm for LS Max $c$-Cut with running time $\mathcal{O}((3eΔ)^k\cdot c\cdot k^3\cdotΔ\cdot n)$, where $Δ$ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for a state-of-the-art heuristic for Max $c$-Cut. We show that using parameterized local search, the results of this state-of-the-art heuristic can be further improved on a set of standard benchmark instances.

cs.CC↗

arXiv2022

The Parameterized Complexity of s-Club with Triangle and Seed Constraints

The s-Club problem asks, for a given undirected graph $G$, whether $G$ contains a vertex set $S$ of size at least $k$ such that $G[S]$, the subgraph of $G$ induced by $S$, has diameter at most $s$. We consider variants of $s$-Club where one additionally demands that each vertex of $G[S]$ is contained in at least $\ell$ triangles in $G[S]$, that each edge of $G[S]$ is contained in at least $\ell$~triangles in $G[S]$, or that $S$ contains a given set $W$ of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size $k$, making them significantly harder than the unconstrained $s$-Club problem. On the positive side, we obtain some FPT algorithms for the case when $\ell=1$ and for the case when $G[W]$, the graph induced by the set of seed vertices, is a clique.

cs.DS↗