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Tutorial: Multiway Cut Problem

1. Problem description:
Given: Ask for:
2. Example:
3. NP-hardness of the problem.

The problem of finding a minimum weight multiway cut is NP-hard for any fixed k ≥ 3.
When k=2, it is the minimum s-t cut problem which can be solved in polynomial time using Maximum Flow algorithm.

4. A 2-2/k approximation algorithm:
  1. For each i = 1, 2, ..., k, compute a minimum weight isolating cut for si , say Ci .
  2. C = C1 ∪ C2 ∪ ... ∪ Ck - Cj in which Cj is the cut with the maximum weight.
  3. Output C.
What's isolating cut? This can be done by identifying the terminals in S-{si} into a single node, and finding a minimum cut separating this node from si, this takes one max-flow computation.
Notice that C contains k-1 sets of isolating cut and the removal of C from the graph G disconnects every pair of terminals, so C is a multiway cut.

Analysis of the approximation ratio: 2-2/k
          i=1k c(Ai) = 2c(A) = 2 • OPT 
          c(Ci) ≤ c(Ai) 
          c(Cj) ≥ 1/k  c(Ci) 
          c(C) ≤ (1-1/k) i=1k c(Ci) ≤ (1-1/k) i=1k c(Ai) ≤ (2-2/k) OPT 

An example showing this ratio is tight:

The blue cut is the approximation solution, c(C) = 6-3ε
The red cut is the OPT solution, c(A) = 4

5. An 3/2-1/k approximation algorithm:
This is an interesting relaxation of linear program by Gruia Calinescu, et al.


            {x ∈ Rk | x ≥ 0 and i=1k xi = 1} .
           x3 = (0,0,1,0,0,0,0); x7 = (0,0,0,0,0,0,1).
              d(u,v)=1/2 t=1k | xut - xvt | 
             e.g.  d(u,s1) = 1/2(|0.5-1|+|0.5-0|+|0-0|) = 0.5  in the following figure.

Example of G with 3 terminals {s1, s2, s3} and 3 non-terminals {u, v, w}:


In this example, the optimal fractional solution has cost 7.5:

How to partition node u, v and w?
1.We need normalization before present the algorithm:

2.We need some definitions before present the algorithm: 3.The algorithm:
  1. Compute the optimal, normalized fractional solution x of LP;
  2. Relabel the terminals so that Wk is the largest among all Wi;
  3. Pick random σ in two permutations: {(1,2,...,k-1,k), (k-1,k-2,...,1,k)};
  4. Pick random ρ∈[0,1];
  5. For i = 1 to k-1 do Vσ(i)=B(i,ρ)-∪j<iVσ(j);
  6. Vk = V - ∪i<k Vi;
  7. Return C = {(u,v) ∈ E | u∈ Vi, v∈ Vj, i ¬= j} .

Analysis of the approximation ratio: 3/2 - 1/k
1. Intuition:


Every color represents a choice of ρ, only when the color line falls into interval (u,v), the edge (u,v) is cut by partition v into S1, partition u into S3.
This gives us probability d(e).


With the same ρ but different σ:

The summation of previous d(e) and d(e)/2 gives us the approximation ratio 3/2. And since the heaviest cut is discarded, we get 3/2-1/k.

2. Formal analysis:

   (1) If  e ∈ E - Ek , Prob[e∈ C]≤ 1.5• d(e) 

   (2) If  e ∈ Ek , Prob[e∈ C]≤ d(e) 
                      Pr[(u,v) ∈ C] = |xui - xvi| = d(e)

   (3) From the analysis of (1) and (2):
E[c(C)] = e∈ E c(e)Pr[e∈ C] = e∈ E-Ek c(e)Pr[e∈ C] + e∈ Ek c(e)Pr[e∈ C] ≤ 1.5 e∈ E-Ek c(e)d(e) + e∈ Ek c(e)d(e) = 1.5 e∈ E c(e)d(e) - 0.5 e∈ Ek c(e)d(e) ≤ (1.5-1/k) • OPTf



[1] Vijay V. Vazirani Approximation Aglorithms, Springer, 2001.
[2] E. Dahlhaus, et al. The Complexity of Multiway Cuts, Extended abstract, 1983.
[3] Gruia Calinescu, et al. An Improved Approximation Algorithm for Multiway Cut, Journal of Computer and System Sciences, 60(3):564-574, 2000.

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