Karger's algorithm

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A graph and two of its cuts. The dotted line in red is a cut with three crossing edges. The dashed line in green is a min-cut of this graph, crossing only two edges.

In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first published in 1993.[1]

The idea of the algorithm is based on the concept of contraction of an edge in an undirected graph . Informally speaking, the contraction of an edge merges the nodes and into one, reducing the total number of nodes of the graph by one. All other edges connecting either or are "reattached" to the merged node, effectively producing a multigraph. Karger's basic algorithm iteratively contracts randomly chosen edges until only two nodes remain; those nodes represent a cut in the original graph. By iterating this basic algorithm a sufficient number of times, a minimum cut can be found with high probability.

The global minimum cut problem[edit]

Main article: Minimum cut

A cut in an undirected graph is a partition of the vertices into two non-empty, disjoint sets . The cutset of a cut consists of the edges between the two parts. The size (or weight) of a cut in an unweighted graph is the cardinality of the cutset, i.e., the number of edges between the two parts,

There are ways of choosing for each vertex whether it belongs to or to , but two of these choices make or empty and do not give rise to cuts. Among the remaining choices, swapping the roles of and does not change the cut, so each cut is counted twice; therefore, there are distinct cuts. The minimum cut problem is to find a cut of smallest size among these cuts.

For weighted graphs with positive edge weights the weight of the cut is the sum of the weights of edges between vertices in each part

which agrees with the unweighted definition for .

A cut is sometimes called a “global cut” to distinguish it from an “- cut” for a given pair of vertices, which has the additional requirement that and . Every global cut is an - cut for some . Thus, the minimum cut problem can be solved in polynomial time by iterating over all choices of and solving the resulting minimum - cut problem using the max-flow min-cut theorem and a polynomial time algorithm for maximum flow, such as the push-relabel algorithm, though this approach is not optimal. Better deterministic algorithms for the global minimum cut problem include the Stoer–Wagner algorithm, which has a running time of .[2]

Contraction algorithm[edit]

The fundamental operation of Karger’s algorithm is a form of edge contraction. The result of contracting the edge is new node . Every edge or for to the endpoints of the contracted edge is replaced by an edge to the new node. Finally, the contracted nodes and with all their incident edges are removed. In particular, the resulting graph contains no self-loops. The result of contracting edge is denoted .

The marked edge is contracted into a single node.

The contraction algorithm repeatedly contracts random edges in the graph, until only two nodes remain, at which point there is only a single cut.

Successful run of Karger’s algorithm on a 10-vertex graph. The minimum cut has size 3.
   procedure contract():
   while 
       choose  uniformly at random
       
   return the only cut in 

When the graph is represented using adjacency lists or an adjacency matrix, a single edge contraction operation can be implemented with a linear number of updates to the data structure, for a total running time of . Alternatively, the procedure can be viewed as an execution of Kruskal’s algorithm for constructing the minimum spanning tree in a graph where the edges have weights according to a random permutation . Removing the heaviest edge of this tree results in two components that describe a cut. In this way, the contraction procedure can be implemented like Kruskal’s algorithm in time .

The random edge choices in Karger’s algorithm correspond to an execution of Kruskal’s algorithm on a graph with random edge ranks until only two components remain.

The best known implementations use time and space, or time and space, respectively.[1]

Success probability of the contraction algorithm[edit]

In a graph with vertices, the contraction algorithm returns a minimum cut with polynomially small probability . Every graph has cuts,[3] among which at most can be minimum cuts. Therefore, the success probability for this algorithm is much better than the probability for picking a cut at random, which is at most

For instance, the cycle graph on vertices has exactly minimum cuts, given by every choice of 2 edges. The contraction procedure finds each of these with equal probability.

To establish the bound on the success probability in general, let denote the edges of a specific minimum cut of size . The contraction algorithm returns if none of the random edges belongs to the cutset of . In particular, the first edge contraction avoids , which happens with probability . The minimum degree of is at least (otherwise a minimum degree vertex would induce a smaller cut), so . Thus, the probability that the contraction algorithm picks an edge from is

The probability that the contraction algorithm on an -vertex graph avoids satisfies the recurrence , with , which can be expanded as

Repeating the contraction algorithm[edit]

10 repetitions of the contraction procedure. The 5th repetition finds the minimum cut of size 3.

By repeating the contraction algorithm times with independent random choices and returning the smallest cut, the probability of not finding a minimum cut is

The total running time for repetitions for a graph with vertices and edges is .

Karger–Stein algorithm[edit]

An extension of Karger’s algorithm due to David Karger and Clifford Stein achieves an order of magnitude improvement.[4]

The basic idea is to perform the contraction procedure until the graph reaches vertices.

   procedure contract(, ):
   while 
       choose  uniformly at random
       
   return 

The probability that this contraction procedure avoids a specific cut in an -vertex graph is

This expression is and becomes less than around . In particular, the probability that an edge from is contracted grows towards the end. This motivates the idea of switching to a slower algorithm after a certain number of contraction steps.

   procedure fastmincut():
   if :
       return mincut()
   else:
       
        contract(, )
        contract(, )
       return min {fastmincut(), fastmincut()}

Analysis[edit]

The probability the algorithm finds a specific cutset is given by the recurrence relation

with solution . The running time of fastmincut satisfies

with solution . To achieve error probability , the algorithm can be repeated times, for an overall running time of . This is an order of magnitude improvement over Karger’s original algorithm.

Finding all min-cuts[edit]

Theorem: With high probability we can find all min cuts in the running time of .

Proof: We know that , therefore after running this algorithm times the probability of missing a specific min-cut is

.

And there are at most min-cuts, hence the probability of missing any min-cut is

The probability of failures is considerably small when n is large enough.∎

Improvement bound[edit]

To determine a min-cut, one has to touch every edge in the graph at least once, which is time in a dense graph. The Karger–Stein's min-cut algorithm takes the running time of , which is very close to that.

References[edit]

  1. ^ a b Karger, David (1993). "Global Min-cuts in RNC and Other Ramifications of a Simple Mincut Algorithm". Proc. 4th Annual ACM-SIAM Symposium on Discrete Algorithms. 
  2. ^ Stoer, M.; Wagner, F. (1997). "A simple min-cut algorithm". Journal of the ACM 44 (4): 585. doi:10.1145/263867.263872. 
  3. ^ Patrignani, Maurizio; Pizzonia, Maurizio (2001), "The complexity of the matching-cut problem", in Brandstädt, Andreas; Le, Van Bang, Graph-Theoretic Concepts in Computer Science: 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14ÔÇô16, 2001, Proceedings, Lecture Notes in Computer Science 2204, Berlin: Springer, pp. 284–295, doi:10.1007/3-540-45477-2_26, MR 1905640 .
  4. ^ Karger, David R.; Stein, Clifford (1996). "A new approach to the minimum cut problem" (PDF). Journal of the ACM 43 (4): 601. doi:10.1145/234533.234534.