The Solution of Some Random NP-Hard Problems in Polynomial Expected Time

Abstract The average-case complexity of recognising some NP-complete properties is examined, when the instances are randomly selected from those which have the property. We carry out this analysis for 1. (1) Graph k -colourability. We describe an O ( n 2 ) expected time algorithm for n -vertex graphs, with k constant. 2. (2) Small equitable cut. We describe an O ( n 3 ) expected time algorithm for finding and verifying , the minimum equitable cut in a 2 n -vertex graph G , condition on G having one with at most (1 − ϵ)n 2 2 edges. 3. (3) Partitioning a 2 n vertex graph into two sparse vertex induced subgraphs of a given class. We describe an O ( n 3 ) expected time algorithm for computing such a partition. 4. (4) The number problem 3-PARTITION. We describe an O ( n 2 ) expected time algorithm for problems with 3 n integers.