Design of Approximation Algorithms

12.4 Everything at once: finding a large cut in a dense graph 323In this section, we will show that we can obtain a PTAS for the unweighted maximum cutproblem in dense graphs by using a sophisticated combination of the randomization techniquesintroduced in the Chapter 5. Recall that a graph is dense if it has at least α ( n2)edges for someconstant α > 0. In Theorem 5.3 of Section 5.1, we gave a simple 1 2-approximation algorithm forthe maximum cut problem. The analysis shows that the algorithm finds a cut whose expectedvalue is at least 1 ∑2 (i,j)∈E w ij. Thus it must be the case that OPT ≥ 1 ∑2 (i,j)∈E w ij. It followsthat in an unweighted dense graph, we know that OPT ≥ α ( n2 2).Recall that in Section 5.12 we introduced a sampling technique for coloring dense 3-colorablegraphs. We would like to use the same sampling technique for the maximum cut problem onunweighted dense graphs. That is, suppose we can draw a sample of the vertices of the graphand assume that we know whether each vertex of the sample is in U or W for an optimal cut.If the sample size is O(log n) we can enumerate all the possible placements of these vertices inU and W , including the one corresponding to an optimal cut. In the case of trying to color a3-colorable graph, a knowledge of the correct coloring of the sample was enough to infer thecoloring of the rest of the graph. What can we do in this case? In the case of coloring, weshowed that with high probability, each vertex in the graph had some neighbor in the sampleS. Here we will show that by using the sample we can get an estimate for each vertex of howmany neighbors are in U in an optimal solution that is accurate to within ±ϵn. Once we havesuch estimates we can use randomized rounding of a linear program in order to determine whichof the remaining vertices should be placed in U. Finally, we use Chernoff bounds to show thatthe solution obtained by randomized rounding is close to the optimal solution.We draw our sample in a slightly different fashion than we did for the 3-coloring algorithm.Given a constant c > 0 and a constant ϵ, 0 < ϵ < 1, we draw a multiset S of exactly (c log n)/ϵ 2vertices by choosing vertices at random with replacement. As in the case of 3-coloring a graphwe can now in polynomial time enumerate all possible ways of splitting the sample set S intotwo parts. Let us say that x i = 0 if we assign vertex i to U and x i = 1 if we assign vertex i toW . Let x ∗ be an optimal solution to the maximum cut problem. Let u i (x) be the number ofneighbors of vertex i in U given an assignment x of vertices. Observe that ∑ ni=1 u i(x)x i givesthe value of cut for the assignment x: when x i = 1 and i ∈ W , there are u i (x) edges from ito vertices in U, so that this sum counts exactly the set of edges in the cut. We can give areasonably good estimate of u i (x) for all vertices i by calculating the number of neighbors of ithat are in S and assigned to U, then scaling by n/|S|. In other words, if û i (x) is our estimateof the neighbors of i in U, thenû i (x) = n ∑(1 − x j ).|S|j∈S:(i,j)∈ENote that we can calculate this estimate given only the values of the x j for j ∈ S.To prove that these estimates are good, we will need the following inequality, known asHoeffding’s inequality.Fact 12.13 (Hoeffding’s inequality): Let X 1 , X 2 , . . . , X l be l independent 0-1 random variables,not necessarily identically distributed. Then for X = ∑ li=1 X i and µ = E[X], and b > 0,thenPr[|X − µ| ≥ b] ≤ e −2b2 /l .We can now prove bounds on the quality of the estimates.Lemma 12.14: With probability 1 − 2n −2c ,u i (x) − ϵn ≤ û i (x) ≤ u i (x) + ϵnElectronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press