Spider Covers and Their Applications

Algorithm 2

The algorithm for the Spider Cover Problem on 𝐺, 𝑠, and 𝒟.

Algorithm  SP_COV(𝐺,𝑠,𝒟)

Set Δ=𝒟   [Family of groups that need to be covered ]

Set 𝑆=∅, 𝑤=0

Repeat

  (i) Compute the minimum integer 𝑤 (with 1≤𝑤≤|Δ|) such that the algorithm

   MAX_SP (𝐻_𝑤,𝑧,𝑛_𝑤(Δ)) outputs a spider 𝑆 in 𝐻_𝑤 reaching a set 𝑋 for which

    𝐶(𝑋)=|{𝐷∈𝑛_𝑤(Δ)∶𝐷∩𝑋≠∅}|≥|𝑛_𝑤(Δ)|/2=|Δ|/2

  (ii) From the spider 𝑆 in 𝐻_𝑤 obtain (via Lemma  4) 𝑤 new spiders in 𝐺 that cover

   at least |Δ|/2 elements of Δ

  (iii) Let Δ be the new family of uncovered subsets, put in 𝒮 the new 𝑤 spiders,

   set 𝑤=𝑤+𝑤.

Until  Δ=∅.

Output:  𝒮 and 𝑤.