arXiv subjects

Nina Klobas

Publications and source records attributed to Nina Klobas.

5 recordsLinked to original sources

arXiv2024

Realizing temporal graphs from fastest travel times

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an $n \times n$ matrix $D$ and a $Δ\in \mathbb{N}$, the goal is to construct a $Δ$-periodic temporal graph with $n$ vertices such that the duration of a fastest path from $v_i$ to $v_j$ is equal to $D_{i,j}$, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs have been well studied and understood since the 1960's (e.g. [Erdős and Gallai, 1960], [Hakimi and Yau, 1965]). As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i.e. non-temporal) counterpart. First, we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the ``tree-likeness''. We prove a tight classification between such parameters that allow fixed-parameter tractability (FPT) and those which imply W[1]-hardness. We show that our problem is W[1]-hard when parameterized by the feedback vertex number (and therefore also any smaller parameter such as treewidth, degeneracy, and cliquewidth) of the underlying graph, while we show that it is in FPT when parameterized by the feedback edge number (and therefore also any larger parameter such as maximum leaf number) of the underlying graph.

cs.DS↗

arXiv2024

The Complexity of Temporal Vertex Cover in Small-Degree Graphs

Temporal graphs naturally model graphs whose underlying topology changes over time. Recently, the problems TEMPORAL VERTEX COVER (or TVC) and SLIDING-WINDOW TEMPORAL VERTEX COVER(or $Δ$-TVC for time-windows of a fixed-length $Δ$) have been established as natural extensions of the classic problem VERTEX COVER on static graphs with connections to areas such as surveillance in sensor networks. In this paper we initiate a systematic study of the complexity of TVC and $Δ$-TVC on sparse graphs. Our main result shows that for every $Δ\geq 2$, $Δ$-TVC is NP-hard even when the underlying topology is described by a path or a cycle. This resolves an open problem from literature and shows a surprising contrast between $Δ$-TVC and TVC for which we provide a polynomial-time algorithm in the same setting. To circumvent this hardness, we present a number of exact and approximation algorithms for temporal graphs whose underlying topologies are given by a path, that have bounded vertex degree in every time step, or that admit a small-sized temporal vertex cover.

cs.DS↗

arXiv2022

The complexity of computing optimum labelings for temporal connectivity

A graph is temporally connected if there exists a strict temporal path, i.e. a path whose edges have strictly increasing labels, from every vertex $u$ to every other vertex $v$. In this paper we study temporal design problems for undirected temporally connected graphs. The basic setting of these optimization problems is as follows: given a connected undirected graph $G$, what is the smallest number $|λ|$ of time-labels that we need to add to the edges of $G$ such that the resulting temporal graph $(G,λ)$ is temporally connected? As it turns out, this basic problem, called MINIMUM LABELING (ML), can be optimally solved in polynomial time. However, exploiting the temporal dimension, the problem becomes more interesting and meaningful in its following variations, which we investigate in this paper. First we consider the problem MIN. AGED LABELING (MAL) of temporally connecting the graph when we are given an upper-bound on the allowed age (i.e. maximum label) of the obtained temporal graph $(G,λ)$. Second we consider the problem MIN. STEINER LABELING (MSL), where the aim is now to have a temporal path between any pair of "terminals" vertices which lie in a subset $R\subseteq V$. This relaxed problem resembles STEINER TREE in static graphs. However, due to the requirement of strictly increasing labels in a temporal path, STEINER TREE is not a special case of MSL. Finally we consider the age-restricted version of MSL, namely MIN. AGED STEINER LABELING (MASL). Our main results are threefold: we prove that (i) MAL becomes NP-complete on undirected graphs, while (ii) MASL becomes W[1]-hard with respect to the number $|R|$ of terminals. On the other hand we prove that (iii) although the age-unrestricted problem MSL is NP-hard, it is in FPT with respect to the number $|R|$ of terminals. That is, adding the age restriction, makes the above problems strictly harder.

cs.DS↗

arXiv2021

Interference-free Walks in Time: Temporally Disjoint Paths

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.

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arXiv2020

Fast recognition of some parametric graph families

We identify all $[1, λ, 8]$-cycle regular $I$-graphs and all $[1, λ, 8]$-cycle regular double generalized Petersen graphs. As a consequence we describe linear recognition algorithms for these graph families. Using structural properties of folded cubes we devise a $o(N \log N)$ recognition algorithm for them. We also study their $[1,λ,4]$, $[1,λ,6]$ and $[2, λ, 6]$-cycle regularity and settle the value of parameter $λ$.

math.CO↗