Finding common structured patterns in linear graphs

A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting ({square subset}) or crossing ({between}). Given a family of linear graphs, and a non-empty subset R ⊆ {<, {square subset}, {between}}, 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 {<, {between}}-structured patterns and {{square subset}, {between}}-structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2 ℋ (k) for {<, {between}}-structured patterns, (ii) k^{1 / 2} for {{square subset}, {between}}-structured patterns, and (iii) O (sqrt(k log k)) for {<, {square subset}, {between}}-structured patterns, where k is the size of the optimal solution and ℋ (k) = ∑_{i = 1}^k 1 / i is the kth harmonic number. Also, we provide combinatorial results concerning different types of structured patterns that are of independent interest in their own right.