By A.I. Mal'cev
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Additional resources for Algorithms and Recursive Functions
For the purpose of proving the desired approximation factor, the value W of the selection constraint is chosen very conservatively. In random instances, slight relaxations still result in feasible solutions for most situations. Thus, we implemented an additional binary search to obtain an appropriate bound W . This adapted bound admits better results which are still feasible. Furthermore we implemented the so-called MinLoss and MaxLoss heuristics, which are the simplest greedy algorithms for the problem of approximating the capacity-maximization problem with a ﬁxed power assignment.
Let (p, q) be an edge of UDG(P ). We show that G contains a path from p to q consisting of at most 9 edges. Let Cp and Cq be the cells of G, such that p ∈ Cp and q ∈ Cq . We distinguish between the three cases listed in Lemma 1. Full Full Full Full p Full Full Non-full q Non-full Non-full q p p q (a) (b) (c) Fig. 6. t. hop distance (i) Consider ﬁrst the case where Cp and Cq are both full. Then, by Lemma 1, either Cp = Cq or Cp and Cq are neighbors. Notice that p is either a hub point of Cp , or it is connected to one by a single edge; and the same holds for q and Cq .
Then, any path in UDG(P ) that begins at a point in C and exits the block of C, must pass through a full cell in C’s block (not including C itself, which may or may not be full). In particular, if there are points of P outside C’s block, then at least one of C’s neighbors is full. pk pk Full C1 C2 C C3 p0 pi Full C1 C2 pi (a) p0 C3 C (b) Fig. 5. Proposition 1 Proof. Let Π = p0 , p1 , · · · , pk be a path that begins at a point p0 ∈ C and exits C’s block, where pk is the ﬁrst point in Π that is not in C’s block.