Cliques, colouring and satisfiability : from structure to algorithms
Christopher Purcell · Warwick Research Archive Portal (University of Warwick) · 2013
We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem.