VertexCoverByDuality

ClassS04CS141 | recent changes | Preferences

Difference (from prior major revision) (no other diffs)

Changed: 20c20,21
an edge packing y is maximal if there is no other feasible edge packing y' such that y \le y'
An edge packing y is maximal if there is no other feasible edge packing y' such that y ≤ y'.
(In other words, no edge weight can be increased without violating some constraint in the dual.)

Changed: 31c32
* * [http://www.cs.ucr.edu/~neal/index.cgi?b=abstracts#Khuller94Primal
* [http://www.cs.ucr.edu/~neal/index.cgi?b=abstracts#Khuller94Primal

Example: min-weight fractional vertex cover

Minimize v c(v) x(v) subject to:
x(v) ≥ 0 for each vertex v ∈ V.
x(u)+x(v) ≥ 1 for each edge (u,v) ∈ E.

Dual: maximum edge packing:

Maximize e y(e) subject to
y(e) ≥ 0 for each edge ∈ E.
e ∋ w y(e) ≤ c(v) for each vertex w.

Using duality to bound OPT integer solution:

cost(OPT vertex cover) ≥ cost(any feasible solution to the dual)
(follows from weak duality.)

linear-time 2-approximation algorithm for min-cost vertex cover using dual

An edge packing y is maximal if there is no other feasible edge packing y' such that y ≤ y'. (In other words, no edge weight can be increased without violating some constraint in the dual.)

algorithm

  1. find a maximal feasible edge packing y
  2. return all vertices v whose packing constraint is tight

prove: the algorithm is a 2-approximation algorithm


References:

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Last edited January 24, 2004 3:09 pm by NealYoung (diff)
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