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2.1 Grimmett-Mcdiarmid's Greedy Algorithm to Find Cliques of Size

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2.1 Grimmett-Mcdiarmid's Greedy Algorithm to Find Cliques of Size S&DS 684: Statistical Inference on Graphs Fall 2018 Lecture 2: The Maximum Clique Problem and the Planted Clique Problem Lecturer: Yihong Wu Scribe: Zifan Li, Sep 5, 2018 [Ed. Aug 23] 1 1 Let !(G(n; 2 )) denotes the maximum size of cliques in G(n; 2 ) which is a random variable. Recall 1 that !(G(n; 2 )) concentrates around 2 log2 n as shown in the previous lecture. In fact, not only there exists a clique of size (2 − ) log2 n, there exist an abundance of them. The reason is that ( n (k) 0 if k = (2 + ) log2 n E[# of cliques of size k] = 2 2 ! : k +1 if k = (2 − ) log2 n Now that the statistical aspect of the problem has been understood, what about the computational aspect? The complexity of the exhaustive search is n ≈ nlog n log2 n and grows superpolynomially in n. What if we limit ourselves to “efficient" algorithms that run in time polynomial in the size of the graph, say, nC for some constant C. In the following section, we are going to present a greedy algorithm that runs in polynomial time (in fact, sublinear time) and is able to find cliques of size (1 − ) log2 n (a factor-of-two approximation of the maximum clique). In contrast, for the max clique problem (finding the maximum clique in a given graph or deciding whether a clique of a given size exists) in the worst case is impossible to approximate even within a factor of n1−, unless P=NP [H˚as99]. This shows the drastic difference between worst-case analsys 1 and average-case analysis, due to the atypicality of the hard instances. Nevertheless, for G(n; 2 ), it remains open whether there exists an efficient algorithm that finds a clique of size bigger than this threshold, say, 1:01 log2 n. 2.1 Grimmett-McDiarmid's greedy algorithm to find cliques of size (1 − ) log2 n Before we present a greedy algorithm that provably works, let us start with another greedy algorithm which is intuitive but might be difficult to analyze. Algorithm 1: Greedy algorithm I Start from an arbitrary vertex Given a clique of size, repeat: Add a vertex randomly from the common neighbors of the existing clique If there is no common neighbors, stop and return the clique The justification to this algorithm is the following: given that we have found an m-clique, for a given vertex v outside, the probability that v is connected to all m vertices in the clique is, assuming 1 that each edge happens with probability 2 independently, −m P(v is connected to all m vertices) = 2 : 1 Therefore, the probability that there exists a v that is connected to all m vertices in the existing clique is −m n −m P(9v connected to all m) = 1 − (1 − 2 ) ! 1 if 2 1=n, e.g., m = (1 − ) log2 n: However, the reasoning is flawed because given the information that we have an existing clique of 1 size m, the probabilities of v connected to each of them is no longer 2 . Furthermore, the edges are not conditionally independent either. Therefore it is unclear how to analyze Algorithm1. Next we present a variant of the greedy algorithm, due to Grimmett-McDiarmid [GM75],1 which is easily analyzable. Algorithm 2: Greedy algorithm II Label the vertices v1; : : : ; vn arbitrarily for t = 1 to n do Given the current clique, if vt is connected to all vertices in the current clique, add vt to the clique. end for Theorem 2.1. Fix any > 0. With probability tending to one as n ! 1, the output of Algorithm 2 is a clique of size at least (1 − ) log2 n. Proof. It is obvious that the output of the above algorithm, denoted as Sn, is a clique. It remains to show that with high probability, the size of the clique is at least (1 − ) log2 n. Let Ti be the time for the size of the clique to grow from i − 1 to i. By the design of the algorithm 2 we can see that Ti's are independent and geometrically distributed as ind −i i Ti ∼ Geo(2 ) ) ETi = 2 : Therefore, P(jSnj ≥ k) = P(T1 + T2 + ··· + Tk ≤ n) k Y n ≥ T ≤ P i k i=1 k Y = 1 − (1 − 2−i)n=k i=1 k ≥ 1 − (1 − 2−k)n=k ≥ 1 − k(1 − 2−k)n=k ! 1 if k = (1 − ) log2 n: 1The original paper [GM75] deals with the number of vertex coloring (so that no adjacent vertices are colored the same) and independent set (subset of vertices that induce an empty graph). Note that cliques in the original graphs 1 1 correspond to independent sets in the complementary graph, and the complement of G(n; 2 ) is still G(n; 2 ). 2This is reminiscent of the coupon collector problem, where it becomes increasingly more difficult to collect the last few uncollected coupons, although here the situation is more drastic. 2 Remark 2.1. The time complexity of Algorithm2 is, in expectation, log2 n X i 2 2 × i = O(n log2 n): i=1 This is sublinear in the size of the graph (Θ(n2) edges). 2.2 Planted Clique Model 2.2.1 Overview The planted clique model is a random graph model which can be described as follows: First choose a subset K of size k uniformly at random from all n vertices and form a clique. The remaining 1 vertex pairs are connected independently with probability 2 . In other words, ( 1 if both i; j 2 K 8i; j P [i ∼ j] = 1 : 2 otherwise 1 1 Denote the resulting graph G ∼ G(n; 2 ; k). Note that G(n; 2 ; 0) is the usual Erd}os-R´enyi graph 1 G(n; 2 ). Ignoring the computation cost, we can use exhaustive search to recover the planted clique with high 1 probability if k ≥ (2 + ) log2 n, because the maximum clique in G(n; 2 ) is approximately 2 log2 n, as shown in Section ??. For the same reason, if k ≤ (2 − ) log2 n, recovering the planted clique 1 is information-theoretically impossible, as there is an abundance of cliques of such size in G(n; 2 ) (cf. Remark ??). However, what if we only consider efficient algorithms? It turns out the state of p the art can only find planted cliques of size k = Ω( n). In the following sections, we will discuss a number of efficient algorithms that is able to recover the p planted clique with high probability if k = Θ( n). p • Degree test method, which works for k = Ω( n log n). We will discuss an iterative version p that works for k = C n for some C > 0, that runs in linear time.3 p • Spectral method, which works for k = C n for some C. This can be improved to arbitrarily small C at the price of time complexity nO(1=C2). p • Semi-definite programming approach, which also work for k = C n. The added advantage is robustness, which the previous methods lack. Remark 2.2. We see that the gap between the information-theoretical limit and what computa- tionally efficient algorithms can achieve is significantly larger for the planted clique problem (log2 n p 1 versus n) than the counterpart for the maximum clique problem in G(n; 2 ) (2 log2 n versus log2 n). 1 Exercise (Slightly smarter exhaustive search). To find the hidden k-clique in G(n; 2 ; k), exhaustive n k Θ(log n) search takes k ∼ n time. Here is an n -time algorithm for all k = Ω(log n). 3Since the graph is dense, here linear time means O(n2). 3 1. By exhaustive search we can find a clique T of size C log2 n for some large constant C. Then jT \ Kj ≥ (C − 2 − ) log2 n. (Why?) 2. Let S denote the set of all vertices that has at least C=2 log n neighbors in T . Then one can show that S ⊂ K with high probability. 3. Now S might also contain some non-clique vertices, which requires some cleanup. So we report the k highest-degree vertices in the induced subgraph G[S]. 2.2.2 Degree Test By degree test we meant declaring the k vertices with the largest degrees as the estimated clique. The motivation is that the vertices in the clique tend to have a higher degree than those outside. To analyze this method, let di denote the degree of a vertex i. If i2 = K, then 1 n d ∼ Binom n − 1; ; d ≈ : i 2 E i 2 If i 2 K, then 1 n + k d ∼ k − 1 + Binom n − k; ; d ≈ : i 2 E i 2 Therefore, the separation in mean is proporational to k. Usually, we need the separation to be at p least as large as the standard deviation. Therefore, k = Ω( pn) is clearly necessary. Furthermore, for the degree test method to work, we need an additional log n factor to accommodate for the possibility that some of the n − k non-clique vertices will have an atypically high degree, and some of the k vertices in the clique will have an atypically low degree. This will be carried out by an union bound, as we shall see later. In fact, although the degrees are not mutually independentp (di and dj are positively correlated through 1 fi ∼ jg), they are almost independent so log n factor is necessary in order for the degrees to fully separate.
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