Set packing is a classical
NP-completeIn computational complexity theory, the complexity class NP-complete , is a class of problems having two properties...
problem in
computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
and
combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
, and was one of
Karp's 21 NP-complete problemsOne of the most important results in computational complexity theory was Stephen Cook's 1971 demonstration of the first NP-complete problem, the boolean satisfiability problem...
.
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
NPIn computational complexity theory, NP is one of the most fundamental complexity classes.The abbreviation NP refers to "nondeterministic polynomial time"....
since, given
k subsets, we can easily verify that they are pairwise disjoint.
The
optimization versionIn mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. More formally, an optimization problem is a quadruple , where* is a set of instances;...
of the problem,
maximum set packing, asks for the maximum number of pairwise disjoint sets in the list.
Set packing is a classical
NP-completeIn computational complexity theory, the complexity class NP-complete , is a class of problems having two properties...
problem in
computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
and
combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
, and was one of
Karp's 21 NP-complete problemsOne of the most important results in computational complexity theory was Stephen Cook's 1971 demonstration of the first NP-complete problem, the boolean satisfiability problem...
.
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
NPIn computational complexity theory, NP is one of the most fundamental complexity classes.The abbreviation NP refers to "nondeterministic polynomial time"....
since, given
k subsets, we can easily verify that they are pairwise disjoint.
The
optimization versionIn mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. More formally, an optimization problem is a quadruple , where* is a set of instances;...
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 problemPacking problems are one area where mathematics meets puzzles . Many of these problems stem from real-life problems with packing items.In a packing problem, you are given:...
s, and its dual linear program is the
set cover problemThe set covering problem is a classical question in computer science and complexity theory.It is a problem "whose study has led to the development of fundamental techniques for the entire field" of approximation algorithms....
.
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.
Heuristics and related problems
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 problemThe set covering problem is a classical question in computer science and complexity theory.It is a problem "whose study has led to the development of fundamental techniques for the entire field" of approximation algorithms....
. 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 algorithmthumb|280px|right|The greedy algorithm determines the minimum number of US coins to give while [[Change-making problem|making change]]. These are the steps a human would take to emulate a greedy algorithm. The coin of the highest value, less than the remaining change owed, is the local optimum...
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 NP-complete
exact coverIn mathematics, given a collection of subsets of a set X, an exact cover is a subcollection of such that each element in X is contained in exactly one subset in .One says that each element in X is covered by exactly one subset in ....
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
NP-completeIn computational complexity theory, the complexity class NP-complete , is a class of problems having two properties...
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 NP-complete via a reduction from the
clique problemIn computational complexity theory, the clique problem is a graph-theoretic NP-complete problem. The problem was one of Richard Karp's original 21 problems shown NP-complete in his 1972 paper "Reducibility Among Combinatorial Problems"...
.
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 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 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 3-element 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, 3-dimensional matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching to 3-uniform hypergraphs...
, and independent set are special cases of set packing. A maximum-size matching can be found in polynomial time, but finding a largest 3-dimensional matching or a largest independent set is NP-hard.
- Packing in a hypergraph
In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them...
.
External links