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Approximating the kMulticut Problem
"... We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem ..."
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Cited by 19 (1 self)
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We study the kmulticut problem: Given an edgeweighted undirected graph, a set of l pairs of vertices, and a target k ≤ l, find the minimum cost set of edges whose removal disconnects at least k pairs. This generalizes the well known multicut problem, where k = l. We show that the kmulticut problem on trees can be approximated within a factor of 8 3 + ɛ, for any fixed ɛ> 0, and within O(log 2 n log log n) on general graphs, where n is the number of vertices in the graph. For any fixed ɛ> 0, we also obtain a polynomial time algorithm for kmulticut on trees which returns a solution of cost at most (2 + ɛ) · OP T, that separates at least (1 − ɛ) · k pairs, where OP T is the cost of the optimal solution separating k pairs. Our techniques also give a simple 2approximation algorithm for the multicut problem on trees using total unimodularity, matching the best known algorithm [8].
Generalized roof duality and bisubmodular functions
, 2010
"... Consider a convex relaxation ˆ f of a pseudoboolean function f. We say that the relaxation is totally halfintegral if ˆ f(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj, xi = 1 ..."
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Consider a convex relaxation ˆ f of a pseudoboolean function f. We say that the relaxation is totally halfintegral if ˆ f(x) is a polyhedral function with halfintegral extreme points x, and this property is preserved after adding an arbitrary combination of constraints of the form xi = xj, xi = 1 − xj, and xi = γ where γ ∈ {0, 1, 1 2} is a constant. A wellknown example is the roof duality relaxation for quadratic pseudoboolean functions f. We argue that total halfintegrality is a natural requirement for generalizations of roof duality to arbitrary pseudoboolean functions. Our contributions are as follows. First, we provide a complete characterization of totally halfintegral relaxations ˆ f by establishing a onetoone correspondence with bisubmodular functions. Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally halfintegral relaxations and relaxations based on the roof duality.
Rounding to an integral program
 In Proceedings of the 4th International Workshop on Efficient and Experimental Algorithms (WEA’05
, 2005
"... Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is rest ..."
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Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is restricted in some sense, although this property may be well hidden. Our method is a natural extension of the threshold rounding technique. 1
1 Feasible Integral Solutions
"... For ease of presentation, we first recall our optimization model: min x,z X i∼j cijzij + nX j=1 wj0(1 − xj) + nX j=1 wj1xj + λ βX b=1 nX j=1 nX l=1 Hb(j)Hb(l)xjxl − 2 nX j=1 Hb(j)xjĤb ..."
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For ease of presentation, we first recall our optimization model: min x,z X i∼j cijzij + nX j=1 wj0(1 − xj) + nX j=1 wj1xj + λ βX b=1 nX j=1 nX l=1 Hb(j)Hb(l)xjxl − 2 nX j=1 Hb(j)xjĤb