A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting ( N ) or crossing ( C ). Given a family of linear graphs, and a non-empty subset R ⊆ {<, N, C}, we are interested in the Maximum Common Structured Pattern (MCSP) problem: find a maximum size edge-disjoint graph, with edge-pairs all comparable by one of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. The MCSP problem generalizes many structure-comparison and structure-prediction problems that arise in computational molecular biology. We give tight hardness results for the MCSP problem for {<, C }-structured pat- terns and { N, C }-structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2H (k) for {<, C }-structured patterns, (ii) √k for { N, C }-structured patterns, and (iii) O( √(k log k) ) for {<, N, C }-structured patterns, where k is the size of the optimal solution and H (k) is the k-th harmonic number. Also, we provide combinatorial results concerning the different types of structured patterns that are of independent interest in their own right.

Common Structured Patterns in Linear Graphs: Approximation and Combinatorics

RIZZI, ROMEO;
2007

Abstract

A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting ( N ) or crossing ( C ). Given a family of linear graphs, and a non-empty subset R ⊆ {<, N, C}, we are interested in the Maximum Common Structured Pattern (MCSP) problem: find a maximum size edge-disjoint graph, with edge-pairs all comparable by one of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. The MCSP problem generalizes many structure-comparison and structure-prediction problems that arise in computational molecular biology. We give tight hardness results for the MCSP problem for {<, C }-structured pat- terns and { N, C }-structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2H (k) for {<, C }-structured patterns, (ii) √k for { N, C }-structured patterns, and (iii) O( √(k log k) ) for {<, N, C }-structured patterns, where k is the size of the optimal solution and H (k) is the k-th harmonic number. Also, we provide combinatorial results concerning the different types of structured patterns that are of independent interest in their own right.
9783540734369
linear graph; Maximum Common Structured Pattern (MCSP) problem; structure-comparison; structure-prediction; computational molecular biology
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11562/409580
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