Karger's algorithm

A cut ${\displaystyle (S,T)}$ in an undirected graph ${\displaystyle G=(V,E)}$ is a partition of the vertices ${\displaystyle V}$ into two non-empty, disjoint sets ${\displaystyle S\cup T=V}$. The cutset of a cut consists of the edges ${\displaystyle \{\,uv\in E\colon u\in S,v\in T\,\}}$ 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,

${\displaystyle w(S,T)=|\{\,uv\in E\colon u\in S,v\in T\,\}|\,.}$

There are ${\displaystyle 2^{|V|}}$ ways of choosing for each vertex whether it belongs to ${\displaystyle S}$ or to ${\displaystyle T}$, but two of these choices make ${\displaystyle S}$ or ${\displaystyle T}$ empty and do not give rise to cuts. Among the remaining choices, swapping the roles of ${\displaystyle S}$ and ${\displaystyle T}$ does not change the cut, so each cut is counted twice; therefore, there are ${\displaystyle 2^{|V|-1}-1}$ distinct cuts. The minimum cut problem is to find a cut of smallest size among these cuts.

For weighted graphs with positive edge weights ${\displaystyle w\colon E\rightarrow \mathbf {R} ^{+}}$ the weight of the cut is the sum of the weights of edges between vertices in each part

${\displaystyle w(S,T)=\sum _{uv\in E\colon u\in S,v\in T}w(uv)\,,}$

which agrees with the unweighted definition for ${\displaystyle w=1}$.

A cut is sometimes called a “global cut” to distinguish it from an “${\displaystyle s}$-${\displaystyle t}$ cut” for a given pair of vertices, which has the additional requirement that ${\displaystyle s\in S}$ and ${\displaystyle t\in T}$. Every global cut is an ${\displaystyle s}$-${\displaystyle t}$ cut for some ${\displaystyle s,t\in V}$. Thus, the minimum cut problem can be solved in polynomial time by iterating over all choices of ${\displaystyle s,t\in V}$ and solving the resulting minimum ${\displaystyle s}$-${\displaystyle t}$ 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 ${\displaystyle O(mn+n^{2}\log n)}$.[2]

The fundamental operation of Karger’s algorithm is a form of edge contraction. The result of contracting the edge ${\displaystyle e=\{u,v\}}$ is new node ${\displaystyle uv}$. Every edge ${\displaystyle \{w,u\}}$ or ${\displaystyle \{w,v\}}$ for ${\displaystyle w\notin \{u,v\}}$ to the endpoints of the contracted edge is replaced by an edge ${\displaystyle \{w,uv\}}$ to the new node. Finally, the contracted nodes ${\displaystyle u}$ and ${\displaystyle v}$ with all their incident edges are removed. In particular, the resulting graph contains no self-loops. The result of contracting edge ${\displaystyle e}$ is denoted ${\displaystyle G/e}$.

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(${\displaystyle G=(V,E)}$):
   while ${\displaystyle |V|>2}$
       choose ${\displaystyle e\in E}$ uniformly at random
       ${\displaystyle G\leftarrow G/e}$
   return the only cut in ${\displaystyle G}$

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 ${\displaystyle O(|V|^{2})}$. 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 ${\displaystyle w(e_{i})=\pi (i)}$ according to a random permutation ${\displaystyle \pi }$. 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 ${\displaystyle O(|E|\log |E|)}$.

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 ${\displaystyle O(|E|)}$ time and space, or ${\displaystyle O(|E|\log |E|)}$ time and ${\displaystyle O(|V|)}$ space, respectively.[1]

Repeating the contraction algorithm

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

By repeating the contraction algorithm ${\displaystyle T={\binom {n}{2}}\ln n}$ times with independent random choices and returning the smallest cut, the probability of not finding a minimum cut is

${\displaystyle \left[1-{\binom {n}{2}}^{-1}\right]^{T}\leq {\frac {1}{e^{\ln n}}}={\frac {1}{n}}\,.}$

The total running time for ${\displaystyle T}$ repetitions for a graph with ${\displaystyle n}$ vertices and ${\displaystyle m}$ edges is ${\displaystyle O(Tm)=O(n^{2}m\log n)}$.

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 ${\displaystyle t}$ vertices.

   procedure contract(${\displaystyle G=(V,E)}$, ${\displaystyle t}$):
   while ${\displaystyle |V|>t}$
       choose ${\displaystyle e\in E}$ uniformly at random
       ${\displaystyle G\leftarrow G/e}$
   return ${\displaystyle G}$

The probability ${\displaystyle p_{n,t}}$ that this contraction procedure avoids a specific cut ${\displaystyle C}$ in an ${\displaystyle n}$-vertex graph is

${\displaystyle p_{n,t}\geq \prod _{i=0}^{n-t-1}{\Bigl (}1-{\frac {2}{n-i}}{\Bigr )}={\binom {t}{2}}{\Bigg /}{\binom {n}{2}}\,.}$

This expression is approximately ${\displaystyle t^{2}/n^{2}}$ and becomes less than ${\displaystyle {\frac {1}{2}}}$ around ${\displaystyle t=n/{\sqrt {2}}}$. In particular, the probability that an edge from ${\displaystyle C}$ 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(${\displaystyle G=(V,E)}$):
   if ${\displaystyle |V|\leq 6}$:
       return mincut(${\displaystyle V}$)
   else:
       ${\displaystyle t\leftarrow \lceil 1+|V|/{\sqrt {2}}\rceil }$
       ${\displaystyle G_{1}\leftarrow }$ contract(${\displaystyle G}$, ${\displaystyle t}$)
       ${\displaystyle G_{2}\leftarrow }$ contract(${\displaystyle G}$, ${\displaystyle t}$)
       return min {fastmincut(${\displaystyle G_{1}}$), fastmincut(${\displaystyle G_{2}}$)}