Chromatic kernel and its applications

In this paper, we study the following Chromatic kernel ( CK ) problem: given an n -partite graph (called a chromatic correlation graph) G=(V,E) with V=V_1⋃⋯⋃ V_n and each partite set V_i containing a constant number λ of vertices, compute a subgraph G[V_CK] of G with exactly one vertex from each partite set and the maximum number of edges or the maximum total edge weight, if G is edge-weighted (among all such subgraphs). CK is a new problem motivated by several applications and no existing algorithm directly solves it. In this paper, we first show that CK is NP-hard even if λ =2 , and cannot be approximated within a factor of 16/17 unless P = NP. Then, we present a random-sampling-based PTAS for dense CK. As its application, we show that CK can be used to determine the pattern of chromosome associations in the nucleus for a population of cells. We test our approach by using random and real biological data; experimental results suggest that our approach yields near optimal solutions, and significantly outperforms existing approaches.