Inclusion--Exclusion Algorithms for Counting Set Partitions

Given an n-element set U and a family of subsets S ⊆ 2U we show how to count the number of k-partitions S1 ∪ · · · ∪ Sk = U into subsets Si ∈ S in time 2nnO(1). The only assumption on S is that it can be enumerated in time 2nnO(1). In effect we get exact algorithms in time 2nnO(1) for a number of wellstudied partition problems including Domatic Number, Chromatic Number, Bounded Component Spanning Forest, Partition into Hamiltonian Subgraphs, and Bin Packing. If only polynomial space is available, our algorithms run in time 3nnO(1) if membership in S can be decided in polynomial time. For Chromatic Number, we present a version that runs in time O(2.2461n) and polynomial space. For Domatic Number, we present a version that runs in time O(2.8805n). Finally, we present a family of polynomial space approximation algorithms

• Type: Conference
• Year: 2006
• Where: FOCS
• Authors: Andreas Björklund, Thore Husfeldt
• Keywords: Chromatic Number, Domatic Number, FOCS 2006, Polynomial Space, Theoretical Computer Science