 Set packing

Set packing is a classical NPcomplete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NPcomplete problems.
Suppose we have 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 intersect). The problem is clearly in NP since, given k subsets, we can easily verify that they are pairwise disjoint.
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, belongs to the class of packing problems, and its dual linear program is the set cover problem.^{[1]}
Coveringpacking dualities Covering problems Packing problems Minimum set cover Maximum set packing Minimum vertex cover Maximum matching Minimum edge cover Maximum independent set Contents
Example
As a simple example, suppose you're at a convention of foreign ambassadors, each of which speaks English and also various other languages. You want to make an announcement to a group of them, but because you don't trust them, you don't want them to be able to speak among themselves without you being able to understand them. To ensure this, you will choose a group such that no two ambassadors speak the same language, other than English. On the other hand you also want to give your announcement to as many ambassadors as possible.
In this case, the elements of the set are languages other than English, and the subsets are the sets of languages spoken by a particular ambassador. If two sets are disjoint, those two ambassadors share no languages other than English. A maximum set packing will choose the largest possible number of ambassadors under the desired constraint. Although this problem is hard to solve in general, in this example a good heuristic is to choose ambassadors who only speak unusual languages first, so that not too many others are disqualified.
Set packing is one among a family of problems related to covering or partitioning the elements of a set. One closely related problem is the set cover problem. Here, we are also given a set S and a list of sets, but the goal is to determine whether we can choose k sets that together contain every element of S. These sets may overlap. The optimization version finds the minimum number of such sets. The maximum set packing need not cover every possible element.
One advantage of the set packing problem is that even if it's hard for some k, it's not hard to find a k for which it is easy on a particular input. For example, we can use a greedy algorithm where we look for the set which intersects the smallest number of other sets, add it to our solution, and remove the sets it intersects. We continually do this until no sets are left, and we have a set packing of some size, although it may not be the maximum set packing. Although no algorithm can always produce results close to the maximum (see next section), on many practical inputs these heuristics do so.
The NPcomplete exact cover problem, on the other hand, requires every element to be contained in exactly one of the subsets. Finding such an exact cover at all, regardless of size, is an NPcomplete problem. 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 NPcomplete via a reduction from the clique problem.
There is a weighted version of the set cover problem in which each subset is assigned a real weight and it is this weight we wish to maximize. In our example above, we might weight the ambassadors according to the populations of their countries, so that our announcement will reach the most people possible. This seems to make the problem harder, but as we explain below, most known results for the general problem apply to the weighted problem as well.
Complexity
The set packing problem is not only NPcomplete, 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 this well.
However, the problem does have a variant which is more tractable: if we assume no subset exceeds k≥3 elements, the answer can be approximated within a factor of k/2 + ε for any ε > 0; in particular, the problem with 3element sets can be approximated within about 50%. In another more tractable variant, if no element occurs in more than k of the subsets, the answer can be approximated within a factor of k. This is also true for the weighted version.
See also
 Matching, 3dimensional matching, and independent set are special cases of set packing. A maximumsize matching can be found in polynomial time, but finding a largest 3dimensional matching or a largest independent set is NPhard.
 Packing in a hypergraph.
Notes
References
 "set packing". Dictionary of Algorithms and Data Structures, editor Paul E. Black, National Institute of Standards and Technology. Note that the definition here is somewhat different.
 Steven S. Skiena. "Set Packing". The Algorithm Design Manual. Last modified June 2, 1997.
 Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski and Gerhard Woeginger. "Maximum Set Packing". A compendium of NP optimization problems. Last modified March 20, 2000.
 Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NPCompleteness. W.H. Freeman. ISBN 0716710455. A3.1: SP3, pg.221.
 Vazirani, Vijay V. (2001). Approximation Algorithms. SpringerVerlag. ISBN 3540653678.
External links
 [1]: A Pascal program for solving the problem. From Discrete Optimization Algorithms with Pascal Programs by MacIej M. Syslo, ISBN 0132155095.
 Benchmarks with Hidden Optimum Solutions for Set Covering, Set Packing and Winner Determination
 Solving packaging problem in PHP
Categories: Combinatorics
 NPcomplete problems
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