Design of Approximation Algorithms

5.12 Random sampling and coloring dense 3-colorable graphs 133Theorem 5.29: If W ∗ ≥ c ln n for some constant c, then with high probability, the total numberof paths using any edge is at most W ∗ + √ cW ∗ ln n.Proof. For each e ∈ E, define random variables Xe, i where Xe i = 1 if the chosen s i -t i path usesedge e, and Xe i = 0 otherwise. Then the number of paths using edge e is Y e = ∑ ki=1 Xi e. Wewant to bound max e∈E Y e , and show that this is close to the LP value W ∗ . CertainlyE[Y e ] =k∑ ∑x ∗ P = ∑x ∗ P ≤ W ∗ ,i=1 P ∈P i :e∈P P :e∈Pby constraint (5.11) from the LP. For a fixed edge e, the random variables X i e are independent,so we can apply the Chernoff bound of Theorem 5.23. Set δ = √ (c ln n)/W ∗ . Since W ∗ ≥ c ln nby assumption, it follows that δ ≤ 1. Then by Theorem 5.23 and Lemma 5.26 with U = W ∗ ,Pr[Y e ≥ (1 + δ)W ∗ ] < e −W ∗ δ 2 /3 = e −(c ln n)/3 = 1n c/3 .Also (1 + δ)W ∗ = W ∗ + √ cW ∗ ln n. Since there can be at most n 2 edges,[]Pr max Y e ≥ (1 + δ)W ∗ ≤ ∑ Pr[Y e ≥ (1 + δ)W ∗ ]e∈Ee∈E≤ n 2 ·1n c/3 = n2−c/3 .For a constant c ≥ 12, this ensures that the theorem statement fails to hold with probability atmost 1 n 2 , and by increasing c we can make the probability as small as we like.Observe that since W ∗ ≥ c ln n, the theorem above guarantees that the randomized algorithmproduces a solution of no more than 2W ∗ ≤ 2 OPT. However, the algorithm mightproduce a solution considerably closer to optimal if W ∗ ≫ c ln n. We also observe the followingcorollary.Corollary 5.30: If W ∗ ≥ 1, then with high probability, the total number of paths using anyedge is O(log n) · W ∗ .Proof. We repeat the proof above with U = (c ln n)W ∗ and δ = 1.In fact, the statement of the corollary can be sharpened by replacing the O(log n) withO(log n/ log log n) (see Exercise 5.13).To solve the linear program in polynomial time, we show that it is equivalent to a polynomiallysizedlinear program; we leave this as an exercise to the reader (Exercise 5.14, to be precise).5.12 Random sampling and coloring dense 3-colorable graphsIn this section we turn to another application of Chernoff bounds. We consider coloring a δ-dense 3-colorable graph. We say that a graph is dense if for some constant α the number ofedges in the graph is at least α ( n2); in other words, some constant fraction of the edges thatcould exist in the graph do exist. A δ-dense graph is a special case of a dense graph. A graphis δ-dense if every node has at least δn neighbors for some constant δ; that is, every node hassome constant fraction of neighbors it could have. Finally, a graph is k-colorable if each of theElectronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press