TY - GEN

T1 - Parameterization above a multiplicative guarantee

AU - Fomin, Fedor V.

AU - Golovach, Petr A.

AU - Lokshtanov, Daniel

AU - Panolan, Fahad

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Publisher Copyright:
© Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I, k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (at most) k + g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I, k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k·g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant > 0, multiplicative parameterization above g(I)1+ of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth.

AB - Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I, k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (at most) k + g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I, k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k·g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant > 0, multiplicative parameterization above g(I)1+ of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth.

KW - Above-guarantee parameterization

KW - Girth

KW - Parameterized complexity

UR - http://www.scopus.com/inward/record.url?scp=85078027136&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2020.39

DO - 10.4230/LIPIcs.ITCS.2020.39

M3 - Conference contribution

AN - SCOPUS:85078027136

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

A2 - Vidick, Thomas

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

T2 - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

Y2 - 12 January 2020 through 14 January 2020

ER -