Abstract
In the Intervalizing Coloured Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses 2O(n/logn)$2^{\mathcal {O}(n/\log n)}$ time. We also give an O?(2n)$\mathcal {O}^{\ast }(2^{n})$ algorithm for the case that the number of colors is not fixed.
Highlights
The area of exact algorithms for NP-hard problems is an old area in the field of design and analysis of algorithms, and one with many important new Theory Comput Syst (2016) 58:273–286 developments
Given a graph G = (V, E) together with a proper vertex colouring c : V → {1, . . . , k} of G (a colouring c is proper if for all edges {v, w} ∈ E: c(v) = c(w)), one must decide if G is subgraph of a properly coloured interval graph, i.e., can we add edges, such that each edge is between vertices of different colors and the result is an interval graph? The problem has its original motivation in DNA physical mapping [13]
Our second result is an algorithm for INTERVALIZING COLOURED GRAPHS; this algorithm runs in O∗(2n)
Summary
The area of exact algorithms for NP-hard problems is an old area in the field of design and analysis of algorithms, and one with many important new. We consider exact algorithms for the problem called INTERVALIZING COLOURED GRAPHS. This problem is defined in the following way. Our second result is an algorithm for INTERVALIZING COLOURED GRAPHS (with no bound on the number of colors); this algorithm runs in O∗(2n). It is a rather simple dynamic programming algorithm, in Held-Karp style, and is the first exact algorithm for the problem.
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