13globalmincutalgorithm

# 13globalmincutalgorithm - 15.082 and 6.855J April 1 2003...

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1 15.082 and 6.855J April 1, 2003 The Global Min Cut problem

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2 Global Min cut INPUT : A network G = (N, A) OUTPUT: A cut (S, N\S) such that cap(S, N\S) is minimum. Note : We do not assume that there is a source node s and destination node t. Typically, but not always, the network is undirected.
3 Application to the TSP: Traveling Salesman Problem What is a minimum length tour that visits each point?

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4 An integer programming formulation Let x ij = 1 if there is an arc (i,j) in the tour T x ij = 0 otherwise. ± There are two arcs incident to node i ± For every cutset (S, N-S), there are two arcs from S to N-S. 2 for each node ij j x i = (1) ,\ for each node set ij iSjNS x SN ∈∈ ≥≠ (2) (3) for each , ij ji xx i j = 0 1 integer for each , ij ij i j ≤≤ (4)
5 Any integer solution will be a tour S Any solution to (1) and (3) and (4) will be the union of disjoint directed cycles. But 2 or more disjoint cycles violates (2).

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6 More on the formulation Suppose that one has a solution to the linear program satisfying (1), (3) and (4) but relaxing the integrality constraints. ± Separation problem: Either show that all constraints in (2) are satisfied or else determine a violated constraint. ,\ 2 for each node set ij iSjNS x SN ∈∈ ≥≠ Interpretation : each cut has flow at least 2. Note: the separation problem is repeatedly solved by the best TSP solvers.
7 The TSP example Let x be a flow satisfying (1), (3) and (4). 2/3 2/3 1/2 Let G = (N, A) be a graph in which u ij is the value x ij in the LP. Let (S, T) be the minimum global cut in G. The solution x is feasible for the LP if and only if the capacity of the cut is at least 2.

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13globalmincutalgorithm - 15.082 and 6.855J April 1 2003...

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