PRUEBA DE HIPÓTESIS PARA CANAL 11, 12, 13 Y 14

In this section we provide primal-dual approximation algorithms for both uncapac- itated and capacitated b–Matching. The capacities b0_i, c0_ij, for vertices and edges re- spectively, need not be integral for this section. Each edge (i, j) has weight w_ij0 . In the uncapacitated casec0_ij =∞. The formulation LP27 captures the basic constraints which are sufficient for the purposes of this section – we are explicitly writing down a relaxation which omits non-bipartite constraints. The system LP28 is the dual of LP27. These formulations are undirected and we have only one variable for both (i, j) = (j, i). ˆ β = minP (i,j)wij0 xij 1 b0_i P jxij ≤1 ∀i 1 c0_ijxij ≤1 ∀(i, j)∈E xij ≥0 (LP27) ˆ β= minP ipi+ P (i,j)qij pi b0 i + pj b0 j + qij c0 ij ≤w0_ij ∀(i, j)∈E pi, qij ≥0 (LP28)

A simple primal-dual algorithm is provided in Algorithm 29. Observe that we main- tain a feasible primal and a feasible dual solution. Observe that after the update, for any deleted edge we have pi

b0_i + pj b0_j ≥w 0 ij. Definition 7.6.1. Let Υ = P

ipi. Define the increase in pi, pj due to the edge

(i, j) to be thedirect contribution of edge (i, j). If the edge (i, j) replaces e1, . . . , e`

(possibly the last edge is replaced fractionally) then make two copies of the edge e`,

one copy got deleted and the other copy stayed in the solution. Therefore without loss of generality define the indirectcontribution of the edge (i, j) to be thesum of

Algorithm 29Linear time single pass algorithm for capacitated b–Matching

1: We start with all pi = 0. Throughout the algorithm we will maintain the invariant pi ≥2Pjwij0 xij. Assume that we have some hypothetical vertex v which has 0 weight

edges to every other node withbv =∞and xvj=bj for allj. 2: foreach new edge e= (i, j) do

(a) If pi b0 i + pj b0 j

≥w_ij0 then do nothing, otherwise:

(b) Letx= min{c0_ij, b0_i, b0_j}.

(c) Delete the cheapest x (fractionally) edges incident to i(and same for j). In more detail: Order the edges{(i0, j)|xi0_j ≥0}in increasing order of w0

i0_j. Find i(j) such that P

i0_<i(j)x_i(j)j < x and P_i0_≤i(j)x_i0_j ≥ x. Set y_i(j)j ← P

i0_≤i(j)x_i0_j −x. For i0> i(j) keepxi0_j unchanged. Fori0 < i(j) set x_i0_j = 0.

the direct and indirect contributions of the edges e1, . . . , e`.

The direct contribution of any edge (i, j) is at most 4w_ij0 xij (two vertices, each of whose

pi value increases by at most 2w0ijxij). We increased pi, pj only when p_bi0 i + pj b0 j < w 0 ij.

Since pi ≥2P_jwij0 xij (and likewise forpj) before the edges incident on i (and some

of the edges incident to j) were deleted; the direct contribution of the edges deleted when (i, j) was inserted is at most 1₂ 2w_ij0 xij

. To see this, divide (i, j) and the deleted edges infinitesimally; for each infinitesimal copy of (i, j) with xij = ∆. If an

infinitesimal copy of (i, j) causes the deletion of e1, e2 (incident at i, j respectively,

each with the same infinitesimal ∆ij amount as (i, j)) then wij0 ≥2(w

0_(e

1) +w0(e2))

because we deleted the cheapest edges. The direct contribution of these edges is ∆(2w0(e1) + 2w0(e2)). Therefore the direct contribution of the edges deleted by the

insertion of (i, j) is at most 1₂ 2w_ij0 xij

=w_ij0 xij. Inductively, the indirect contribution

of edge (i, j) is also at most 2w0_ijxij using the facts that the weights of the (sets of)

edges in a chain of deletions decrease geometrically by factor 2. Therefore P ipi =

Υ ≤ 6P

(i,j)w

0

ijxij. The above accounting of the charge is the “trail of the dead”

analysis in [43], and can also be found in the analysis of call-admission algorithms [1]. Therefore if at the end we are left with a setS of edges; for the capacitated problem we set qij =wij0 yij for (i, j)∈S and 0 otherwise. This is a feasible dual solution and

observe that P ipi+ P i,jqij ≤ 7 P (i,j)∈Sw 0

ijxij. But this is a feasible dual solution

and therefore ˆβ ≤ 7P

(i,j)∈Sw

0

ijxij. Furthermore, observe that the solution either

Theorem 7.6.2. We can solve the capacitatedb–Matching problem to an approxima- tion factor 7 within the optimum fractional solution. Moreover the number of edges in the solution is min{m, B}.

For the uncapacitated case, the variables qij do not exist. Therefore P_ipi ≤

6P

(i,j)∈Sw

0

ijxij. This gives a 6 approximation and we have at most O(n) edges.

Theorem 7.6.3. We can solve the uncapacitated b–Matching problem to an approx-

imation factor 6 of the optimum fractional solution. Moreover the number of edges in the solution is O(n). Furthermore setting y_ij† = 6xij will give us a solution where P (i,j)x † ijw 0 ij ≥β ∗ _andP jy † ij ≤6b 0

i which gives us the initial solution for Algorithm 27 as well as the solution sought after in Lemma 7.3.2.