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Data Structures and Algorithms

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From arXiv

Fixed-Parameter Complexity of Minimum Profile Problems

Let

G=(V,E)

be a graph. An ordering of

G

is a bijection

\alpha: V\dom \{1,2,..., |V|\}.

For a vertex

v

G

, its closed neighborhood is

N[v]=\{u\in V: uv\in E\}\cup \{v\}.

The profile of an ordering

\alpha

G

\prf_{\alpha}(G)=\sum_{v\in V}(\alpha(v)-\min\{\alpha(u): u\in N[v]\}).

The profile

\prf(G)

G

is the minimum of

\prf_{\alpha}(G)

over all orderings

\alpha

G

. It is well-known that

\prf(G)

is the minimum number of edges in an interval graph

H

that contains

G

is a subgraph. Since

|V|-1

is a tight lower bound for the profile of connected graphs

G=(V,E)

, the parametrization above the guaranteed value

|V|-1

is of particular interest. We show that deciding whether the profile of a connected graph

G=(V,E)

is at most

|V|-1+k

is fixed-parameter tractable with respect to the parameter

k

. We achieve this result by reduction to a problem kernel of linear size.

Simplify

Published on April 24, 2006

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