arXiv subjects

Enna Gerhard

Publications and source records attributed to Enna Gerhard.

3 recordsLinked to original sources

arXiv2026

FO Value Discovery and Partial Vertex Cover Discovery

We study solution discovery in the token-sliding model from a logical and cost-value optimization perspective. In solution discovery, we are given a graph, an initial placement of $k$ tokens, and a movement budget $b$. The task is to find a reachable target configuration satisfying a prescribed condition. Our results are inspired by \textsc{Partial Vertex Cover Discovery}, where the condition is that the~$k$ tokens cover at least $t$ edges of the input graph. This objective is not merely a sum of independent occupied vertex contributions: each selected vertex contributes its degree, but edges with both endpoints selected have to be subtracted once. To capture this phenomenon, we introduce \textsc{FO Value Discovery}, an optimization problem in which the value of a selected tuple is given by unary vertex weights together with first-order definable correction terms. We further generalize the setting to \textsc{FO Cost-Value Decision}, where vertices carry both costs and values, and the task is to decide whether there is a tuple whose first-order value expression reaches a prescribed value threshold while respecting a cost bound. Finally, we study the parameterized complexity of \textsc{Partial Vertex Cover Discovery} and \textsc{Vertex Cover Discovery}. As a consequence of the logical meta-theorems, we obtain fixed-parameter tractability of \textsc{Partial Vertex Cover Discovery} on several graph classes, including classes of locally bounded cliquewidth. We also show that \textsc{Partial Vertex Cover Discovery} is W[1]-hard parameterized by $k+b$ and fixed-parameter tractable on $d$-degenerate graphs parameterized by $k+d$. For \textsc{Vertex Cover Discovery}, we prove NP-hardness on planar graphs, W[1]-hardness parameterized by the clique cover number, even when a clique cover is supplied with the input, and W[1]-hardness with respect to parameter cutwidth.

cs.DM↗

arXiv2026

Separating Feasibility and Movement in Solution Discovery: The Case of Path Discovery

We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through \textsc{Path Discovery} and \textsc{Shortest Path Discovery}, where the task is to realize a vertex set containing an $s$-$t$-path or a shortest $s$-$t$-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.

cs.DM↗

arXiv2023

Data reduction for directed feedback vertex set on graphs without long induced cycles

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 $\Omega(n^d)$. We show how to compute a kernel with at most $2^dk^d$ vertices and at most $d^{3d}k^d$ induced cycles of length at most $d$, 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. We prove that for every nowhere dense class $\mathscr{C}$ there is a function $f_\mathscr{C}(d,\epsilon)$ such that for graphs $G\in \mathscr{C}$ without induced cycles of length greater than $d$ we can compute a kernel with $f_\mathscr{C}(d,\epsilon)\cdot k^{1+\epsilon}$ vertices for any $\epsilon>0$ in time $f_\mathscr{C}(d,\epsilon)\cdot n^{O(1)}$. The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these classes have treewidth $O(d)$ and hence DFVS on planar graphs without cycles of length greater than $d$ can be solved in time $2^{O(d)}\cdot n^{O(1)}$. We finally present a new data reduction rule for general DFVS and prove that the rule together with two standard rules subsumes all rules applied in the work of Bergougnoux et al.\ to obtain a polynomial kernel for DFVS[FVS], i.e., DFVS parameterized by the feedback vertex set number of the underlying (undirected) graph. We conclude by studying the LP-based approximation of DFVS.

cs.DS↗