Set Partitioning via Inclusion-Exclusion

Abstract

Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2 n n O(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform.In effect we get exact algorithms in 2 n n O(1) time for several well-studied partition problems including Domatic Number, Chromatic Number, Maximum k-Cut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering.If only polynomial space is available, our algorithms run in time 3 n n O(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461 n ) time and Domatic Number in O(2.8718 n ) time.Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and (1 + )χ(G) in time O(1.2209 n + 2.2461 e − n ).

Exact Algorithms for Set Multicover and Multiset Multicover Problems

“…By using the inclusion-exclusion principle (Equation 1), Björklund et al [1] prove that the number c k (F sc ) of set covers can be calculated through Equation 2. Here a sc (X) denotes the number of sets in F sc which avoid (do not cover) any element in the set X ⊆ N .…”

“…Table 1), we can see that, in order to find the minimum number of sets that satisfy the coverage requirement, we just need to find the minimum k value that satisfy c k (F sc ) > 0 using binary search. This is a standard approach which was first used in [1]. Hua et al [6] also employed a similar approach for exactly solving the set multicover problem, i.e., searching the minimum k that guarantees a positive c k (F mc ) number of set multicovers.…”

“…If Y (i) = 0 then any (multi)set S ∈ Fmc (Fms) which covers element i will be deleted. 1 We will use Y instead of Y (X) in a clear context.…”

“…By using the inclusion-exclusion principle and a fast zeta transform technique, Björklund et al [1] have shown that the set cover problem can be exactly solved in O * (2 n ) time using O * (2 n ) space. Here, using the O * (f (n)) notation we omit a (log f (n)) O (1) factor.…”

“…Here, using the O * (f (n)) notation we omit a (log f (n)) O (1) factor. Based on this observation, they proposed a family of exact algorithms for the set partitioning problems which improve all the previous algorithms.…”