TY - GEN
T1 - Temporal Graph Realization from Fastest Paths
AU - Klobas, Nina
AU - Mertzios, George B.
AU - Molter, Hendrik
AU - Spirakis, Paul G.
N1 - Publisher Copyright:
© Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis.
PY - 2024/6/1
Y1 - 2024/6/1
N2 - 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 × n matrix D and a ∆ ∈ N, the goal is to construct a ∆-periodic temporal graph with n vertices such that the duration of a fastest path from vi to vj is equal to Di,j, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs has 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.
AB - 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 × n matrix D and a ∆ ∈ N, the goal is to construct a ∆-periodic temporal graph with n vertices such that the duration of a fastest path from vi to vj is equal to Di,j, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs has 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.
KW - Temporal graph
KW - fastest temporal path
KW - graph realization
KW - parameterized complexity
KW - periodic temporal labeling
KW - temporal connectivity
UR - http://www.scopus.com/inward/record.url?scp=85195370389&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SAND.2024.16
DO - 10.4230/LIPIcs.SAND.2024.16
M3 - Conference contribution
AN - SCOPUS:85195370389
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 3rd Symposium on Algorithmic Foundations of Dynamic Networks, SAND 2024
A2 - Casteigts, Arnaud
A2 - Kuhn, Fabian
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 3rd Symposium on Algorithmic Foundations of Dynamic Networks, SAND 2024
Y2 - 5 June 2024 through 7 June 2024
ER -