Жиын жинағының проблемасы: NP-толықтығы және жуықтау алгоритмдері

Set packing is a classical NP complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP complete problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element). More formally, given a universe and a family of subsets of , a packing is a subfamily of sets such that all sets in are pairwise disjoint. The size of the packing is In the set packing decision problem, the input is a pair and an integer ; the question is whether
there is a set packing of size or more. In the set packing optimization problem, the input is a pair , and the task is to find a set packing that uses the most sets. The problem is clearly in NP since, given subsets, we can easily verify that they are pairwise disjoint in polynomial time. The optimization version of the problem, maximum set packing, asks for the maximum number of pairwise disjoint sets in the list. It is a maximization problem that can be formulated naturally as an integer linear program, belonging to the class of packing problems.

The maximum set packing problem can be formulated as the following integer linear program. maximize (maximize the total number of subsets) subject to for all (selected sets have to be pairwise disjoint) for all (every set is either in the set packing or not)

The set packing problem is not only NP complete, but its optimization version (general maximum set packing problem) has been proven as difficult to approximate as the maximum clique problem; in particular, it cannot be approximated within any constant factor. The best known algorithm approximates it within a factor of The weighted variant can also be approximated as well.

The problem does have a variant which is more tractable. Given any positive integer k≥3, the k set packing problem is a variant of set packing in which each set contains at most k elements. When k=1, the problem is trivial. When k=2, the problem is equivalent to finding a maximum cardinality matching, which can be solved in polynomial time. For any k≥3, the problem is NP hard, as it is more general than 3 dimensional matching. However, there are constant factor approximation algorithms:

Cygan presented an algorithm that, for any ε>0, attains a (k+1+ε)/3 approximation. The run time is polynomial in the number of sets and elements, but doubly exponential in 1/ε. Furer and Yu presented an algorithm that attains the same approximation, but with run time singly exponential in 1/ε.

In another more tractable variant, if no element occurs in more than d of the subsets, the answer can be approximated within a factor of d. This is also true for the weighted version.

Graph matching is a special case of set packing in which the size of all sets is 2 (the sets correspond to the edges). In this special case, a maximum size matching can be found in polynomial time. 3 dimensional matching is a special case in which the size of all sets is 3, and in addition, the elements are partitioned into 3 colors and each set contains exactly one element of each color. This special case is still NP hard, though it has better constant factor approximation algorithms than the general case.

In the set cover problem, we are given a family of subsets of a universe , and the goal is to determine whether we can choose t sets that together contain every element of These sets may overlap. The optimization version finds the minimum number of such sets. The maximum set packing need not cover every possible element. In the exact cover problem, every element of should be contained in exactly one of the subsets. Finding such an exact cover is an NP complete problem, even in the special case in which the size of all sets is 3 (this special case is called exact 3 cover or X3C). However, if we create a singleton set for each element of S and add these to the list, the resulting problem is about as easy as set packing. Karp originally showed set packing NP complete via a reduction from the clique problem.