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Solving $$(k-1)$$ ( k - 1 ) -stable instances of k-terminal cut with isolating cuts

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  • Mark Velednitsky

    (University of California)

Abstract

The k-terminal cut problem, also known as the multiterminal cut problem, is defined on an edge-weighted graph with k distinct vertices called “terminals.” The goal is to remove a minimum weight collection of edges from the graph such that there is no path between any pair of terminals. The problem is APX-hard. Isolating cuts are minimum cuts which separate one terminal from the rest. The union of all the isolating cuts, except the largest, is a $$(2-2/k)$$ ( 2 - 2 / k ) -approximation to the optimal k-terminal cut. An instance of k-terminal cut is $$\gamma $$ γ -stable if edges in the cut can be multiplied by up to $$\gamma $$ γ without changing the unique optimal solution. In this paper, we show that, in any $$(k-1)$$ ( k - 1 ) -stable instance of k-terminal cut, the source sets of the isolating cuts are the source sets of the unique optimal solution to that k-terminal cut instance. We conclude that the $$(2-2/k)$$ ( 2 - 2 / k ) -approximation algorithm returns the optimal solution on $$(k-1)$$ ( k - 1 ) -stable instances. Ours is the first result showing that this $$(2-2/k)$$ ( 2 - 2 / k ) -approximation is an exact optimization algorithm on a special class of graphs. We also show that our $$(k-1)$$ ( k - 1 ) -stability result is tight. We construct $$(k-1-\epsilon )$$ ( k - 1 - ϵ ) -stable instances of the k-terminal cut problem which only have trivial isolating cuts: that is, the source set of the isolating cuts for each terminal is just the terminal itself. Thus, the $$(2-2/k)$$ ( 2 - 2 / k ) -approximation does not return an optimal solution.

Suggested Citation

  • Mark Velednitsky, 2022. "Solving $$(k-1)$$ ( k - 1 ) -stable instances of k-terminal cut with isolating cuts," Journal of Combinatorial Optimization, Springer, vol. 43(2), pages 297-311, March.
  • Handle: RePEc:spr:jcomop:v:43:y:2022:i:2:d:10.1007_s10878-021-00769-3
    DOI: 10.1007/s10878-021-00769-3
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    References listed on IDEAS

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    1. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    2. Mark Velednitsky & Dorit S. Hochbaum, 0. "Isolation branching: a branch and bound algorithm for the k-terminal cut problem," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-21.
    3. David R. Karger & Philip Klein & Cliff Stein & Mikkel Thorup & Neal E. Young, 2004. "Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 436-461, August.
    4. Stephen M. Robinson, 1977. "A Characterization of Stability in Linear Programming," Operations Research, INFORMS, vol. 25(3), pages 435-447, June.
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