[Paper Review] Maximizing the Cohesion is NP-hard

Adrien Friggeri, Éric Fleury|arXiv (Cornell University)|Sep 9, 2011

Complex Network Analysis Techniques3 references3 citations

TL;DR

This paper proves that finding a subset of vertices with maximum cohesion in an undirected network is NP-hard by reducing the NP-complete Clique problem to a decision variant called Connected-Cohesive. The authors establish that maximum-cohesion sets must be connected, and through a polynomial-time transformation, they show that a clique of size k exists in the original graph if and only if a connected subset with a specific cohesion threshold exists in the transformed graph.

ABSTRACT

We show that the problem of finding a set with maximum cohesion in an undirected network is NP-hard.

Motivation & Objective

• To establish the computational complexity of finding a set with maximum cohesion in an undirected network.
• To demonstrate that the problem of maximizing cohesion is NP-hard by proving the NP-completeness of the decision variant Connected-Cohesive.
• To show that any set achieving maximum cohesion must be connected, thereby restricting the search space to connected subsets.
• To provide a polynomial-time reduction from the Clique problem to Connected-Cohesive, proving NP-completeness of the decision problem.
• To confirm that the optimization problem of maximizing cohesion is NP-hard as a consequence of the decision problem being NP-complete.

Proposed method

• Define cohesion mathematically as $ \mathcal{C}(S) = \frac{i(S)^2}{\binom{|S|}{3}(i(S) + o(S))} $, where $ i(S) $ is the number of internal triangles and $ o(S) $ the number of outbound triangles in subset $ S $.
• Prove that any maximum-cohesion set must be connected using a contradiction argument based on cohesion comparison between disconnected subsets and their union.
• Introduce the decision problem Connected-Cohesive: given a graph $ G $ and threshold $ \lambda $, does there exist a connected subset $ S $ with $ \mathcal{C}(S) \geq \lambda $?
• Construct a polynomial-time transformation from an instance of the Clique problem to an instance of Connected-Cohesive by adding large cliques to connect non-edges.
• Use the transformation to show that a clique of size $ k $ exists in the original graph if and only if a connected subset with cohesion $ \lambda = \frac{{k\choose 3}}{{k\choose 3} + {k\choose 2}(n-k)} $ exists in the transformed graph.
• Establish that Connected-Cohesive is in NP and NP-hard, thus NP-complete, via the reduction from Clique.

Experimental results

Research questions

• RQ1 Is the problem of finding a subset of vertices with maximum cohesion in an undirected graph computationally tractable?
• RQ2 Can the maximum-cohesion set be restricted to connected subsets without loss of optimality?
• RQ3 Is the decision problem of determining whether a connected subset exists with cohesion at least $ \lambda $ NP-complete?
• RQ4 Can the Clique problem be reduced to the Connected-Cohesive decision problem in polynomial time?
• RQ5 Does the NP-completeness of Connected-Cohesive imply that the optimization problem of maximizing cohesion is NP-hard?

Key findings

• The problem of finding a set with maximum cohesion in an undirected network is NP-hard, as the decision variant Connected-Cohesive is NP-complete.
• Any set achieving maximum cohesion must be connected, which allows restricting the search to connected subsets.
• The decision problem Connected-Cohesive is NP-complete, proven via a polynomial-time reduction from the Clique problem (GT19).
• A clique of size $ k $ exists in the original graph $ G $ if and only if a connected subset with cohesion $ \lambda = \frac{{k\choose 3}}{{k\choose 3} + {k\choose 2}(n-k)} $ exists in the transformed graph $ G' $.
• The transformation from Clique to Connected-Cohesive runs in polynomial time, confirming the NP-hardness of the cohesion maximization problem.
• The result implies that optimizing for cohesion in social and complex networks is computationally intractable in the general case.