TY - GEN

T1 - Quasipolynomial representation of transversal matroids with applications in parameterized complexity

AU - Lokshtanov, Daniel

AU - Misra, Pranabendu

AU - Panolan, Fahad

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Publisher Copyright: © Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh and Meirav Zehavi.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union representation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the transversal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r ∈ N, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union representation. Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element.

AB - Deterministic polynomial-time computation of a representation of a transversal matroid is a longstanding open problem. We present a deterministic computation of a so-called union representation of a transversal matroid in time quasipolynomial in the rank of the matroid. More precisely, we output a collection of linear matroids such that a set is independent in the transversal matroid if and only if it is independent in at least one of them. Our proof directly implies that if one is interested in preserving independent sets of size at most r, for a given r ∈ N, but does not care whether larger independent sets are preserved, then a union representation can be computed deterministically in time quasipolynomial in r. This consequence is of independent interest, and sheds light on the power of union representation. Our main result also has applications in Parameterized Complexity. First, it yields a fast computation of representative sets, and due to our relaxation in the context of r, this computation also extends to (standard) truncations. In turn, this computation enables to efficiently solve various problems, such as subcases of subgraph isomorphism, motif search and packing problems, in the presence of color lists. Such problems have been studied to model scenarios where pairs of elements to be matched may not be identical but only similar, and color lists aim to describe the set of compatible elements associated with each element.

KW - Matroid representation

KW - Representative set

KW - Travserval matroid

KW - Union representation

UR - http://www.scopus.com/inward/record.url?scp=85041662642&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ITCS.2018.32

DO - https://doi.org/10.4230/LIPIcs.ITCS.2018.32

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 9th Innovations in Theoretical Computer Science, ITCS 2018

A2 - Karlin, Anna R.

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

T2 - 9th Innovations in Theoretical Computer Science, ITCS 2018

Y2 - 11 January 2018 through 14 January 2018

ER -