VertexCoverByDuality

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:

• Chapter 15 of Approximation Algorithms by Vazirani
• [A primal-dual parallel approximation technique applied to weighted set and vertex cover] - a parallel/distributed algorithm.