[Paper Review] Maximum Cliques in Graphs with Small Intersection Number and Random Intersection Graphs

Sotiris Nikoletseas, Christoforos Raptopoulos+1 more|arXiv (Cornell University)|Apr 18, 2012

Complexity and Algorithms in Graphs13 references3 citations

TL;DR

This paper presents a polynomial-time algorithm for finding maximum cliques in random intersection graphs when the number of labels is polynomial in the number of vertices and the label selection probability is small. The key contribution is a Single Label Clique Theorem proving that, with high probability, large cliques are formed by a single label, enabling efficient clique detection via label analysis.

ABSTRACT

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of $m$ labels is chosen independently with probability $p$ by each one of $n$ vertices, and there are edges between any vertices with overlaps in the labels chosen). We first present a simple algorithm which, on input $G$ finds a maximum clique in $O(2^{2^m + O(m)} + n^2 \min\{2^m, n\})$ time steps, where $m$ is an upper bound on the intersection number and $n$ is the number of vertices. Consequently, when $m \leq \ln{\ln{n}}$ the running time of this algorithm is polynomial. We then consider random instances of the random intersection graphs model as input graphs. As our main contribution, we prove that, when the number of labels is not too large ($m=n^α, 0< α<1$), we can use the label choices of the vertices to find a maximum clique in polynomial time whp. The proof of correctness for this algorithm relies on our Single Label Clique Theorem, which roughly states that whp a "large enough" clique cannot be formed by more than one label. This theorem generalizes and strengthens other related results in the state of the art, but also broadens the range of values considered. As an important consequence of our Single Label Clique Theorem, we prove that the problem of inferring the complete information of label choices for each vertex from the resulting random intersection graph (i.e. the \emph{label representation of the graph}) is \emph{solvable} whp. Finding efficient algorithms for constructing such a label representation is left as an interesting open problem for future research.

Motivation & Objective

• To address the intractability of the maximum clique problem in general graphs by focusing on graphs with small intersection number.
• To develop efficient algorithms for computing maximum cliques in random intersection graphs, a model relevant to real-world networks.
• To establish conditions under which the maximum clique can be found in polynomial time with high probability.
• To prove that large cliques in random intersection graphs are typically formed by a single label, enabling label-based clique detection.
• To show that the label representation of the graph can be uniquely reconstructed from the graph structure with high probability.

Proposed method

• Proposes a brute-force algorithm that enumerates all subsets of cliques up to size 2^m, leveraging an upper bound m on the intersection number.
• Uses Chernoff bounds to establish concentration of label degrees and vertex label set sizes in random intersection graphs.
• Introduces the Single Label Clique Theorem, proving that cliques of size close to np are unlikely to be formed by more than one label when mp² = O(1).
• Employs probabilistic analysis to bound the probability of multiple-label cliques, showing such cliques are rare under the given parameter regime.
• Applies the maximum likelihood estimation method to reconstruct the label sets of vertices from the graph, proving uniqueness with high probability.
• Analyzes the random intersection graph model G_{n,m,p} with m = n^α, 0 < α < 1, and p such that mp² = O(1), to derive structural properties of cliques.

Experimental results

Research questions

• RQ1 Under what conditions can the maximum clique in a random intersection graph be computed in polynomial time?
• RQ2 Can large cliques in random intersection graphs be shown to be formed predominantly by a single label?
• RQ3 Is the label representation of a random intersection graph uniquely reconstructible from the graph structure with high probability?
• RQ4 How does the intersection number relate to the tractability of the maximum clique problem?
• RQ5 What is the asymptotic size of the maximum clique in random intersection graphs when m = n^α and mp² = O(1)?

Key findings

• When the intersection number m satisfies m ≤ ln ln n, the proposed algorithm runs in polynomial time.
• For random intersection graphs with m = n^α (0 < α < 1) and mp² = O(1), the maximum clique can be found in polynomial time with high probability.
• The size of the maximum clique in G_{n,m,p} with m = n^α and mp² = O(1) is asymptotically np.
• The Single Label Clique Theorem shows that with high probability, no large clique is formed by more than one label, enabling efficient clique detection.
• The label representation of the graph is uniquely reconstructible from the graph structure with high probability using maximum likelihood estimation.
• The chromatic number of G_{n,m,p} in the regime mp = ω(ln n) is Ω(nmp² / ln n), indicating the graph is far from perfect, contrasting with the clique structure in the studied regime.