TY - GEN

T1 - The complexity of finding small separators in temporal graphs

AU - Zschoche, Philipp

AU - Fluschnik, Till

AU - Molter, Hendrik

AU - Niedermeier, Rolf

N1 - Publisher Copyright:
© Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Temporal graphs are graphs with time-stamped edges. We study the problem of finding a small vertex set (the separator) with respect to two designated terminal vertices such that the removal of the set eliminates all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that pass through arbitrarily many edges per time step (non-strict) and paths that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-hardness versus polynomial-time solvability) for both problem variants. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We further show that, on temporal graphs with planar underlying graph, if additionally the number of time steps is constant, then the problem variant for strict paths is solvable in quasi-linear time. Finally, we introduce and motivate the notion of a temporal core (vertices whose incident edges change over time). We prove that the non-strict variant is fixed-parameter tractable when parameterized by the size of the temporal core, while the strict variant remains NP-complete, even for constant-size temporal cores.

AB - Temporal graphs are graphs with time-stamped edges. We study the problem of finding a small vertex set (the separator) with respect to two designated terminal vertices such that the removal of the set eliminates all temporal paths connecting one terminal to the other. Herein, we consider two models of temporal paths: paths that pass through arbitrarily many edges per time step (non-strict) and paths that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-hardness versus polynomial-time solvability) for both problem variants. Moreover we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. We further show that, on temporal graphs with planar underlying graph, if additionally the number of time steps is constant, then the problem variant for strict paths is solvable in quasi-linear time. Finally, we introduce and motivate the notion of a temporal core (vertices whose incident edges change over time). We prove that the non-strict variant is fixed-parameter tractable when parameterized by the size of the temporal core, while the strict variant remains NP-complete, even for constant-size temporal cores.

KW - (non-)strict temporal paths

KW - Length-bounded cuts

KW - Node multiway cut

KW - Parameterized complexity

KW - Single-source shortest paths

KW - Temporal core

UR - http://www.scopus.com/inward/record.url?scp=85053214638&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2018.45

DO - 10.4230/LIPIcs.MFCS.2018.45

M3 - Conference contribution

AN - SCOPUS:85053214638

SN - 9783959770866

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018

A2 - Potapov, Igor

A2 - Worrell, James

A2 - Spirakis, Paul

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018

Y2 - 27 August 2018 through 31 August 2018

ER -