TY - JOUR

T1 - Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques

AU - Kaplan, Haim

AU - Shamir, Ron

PY - 1996/6

Y1 - 1996/6

N2 - We study two related problems motivated by molecular biology. • Given a graph G and a constant fc, does there exist a supergraph G′ of G that is a unit interval graph and has clique size at most k? • Given a graph G and a proper k-coloring c of G, does there exist a supergraph G′ of G that is properly colored by c and is a unit interval graph? We show that those problems are polynomial for fixed k. On the other hand, we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k - 1. Hence, it is NP-hard and W[t]-hard for all t. We also show that the second problem is W[1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(nα) algorithm, where α is a constant independent of k. A central tool in our study is a new graph-theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph.

AB - We study two related problems motivated by molecular biology. • Given a graph G and a constant fc, does there exist a supergraph G′ of G that is a unit interval graph and has clique size at most k? • Given a graph G and a proper k-coloring c of G, does there exist a supergraph G′ of G that is properly colored by c and is a unit interval graph? We show that those problems are polynomial for fixed k. On the other hand, we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k - 1. Hence, it is NP-hard and W[t]-hard for all t. We also show that the second problem is W[1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(nα) algorithm, where α is a constant independent of k. A central tool in our study is a new graph-theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph.

KW - Bandwidth

KW - Design and analysis of algorithms

KW - Interval graphs

KW - Parameterized complexity

KW - Pathwidth

UR - http://www.scopus.com/inward/record.url?scp=0037668746&partnerID=8YFLogxK

U2 - 10.1137/S0097539793258143

DO - 10.1137/S0097539793258143

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AN - SCOPUS:0037668746

SN - 0097-5397

VL - 25

SP - 540

EP - 561

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 3

ER -