Chromatic kernel and its applications

In this paper, we study the following Chromatic kernel (CK) problem: given an $$n$$n-partite graph (called a chromatic correlation graph) $$G=(V,E)$$G=(V,E) with $$V=V_{1}\bigcup \cdots \bigcup V_{n}$$V=V1⋃⋯⋃Vn and each partite set $$V_{i}$$Vi containing a constant number $$\lambda $$λ of vertices, compute a subgraph $$G[V_{CK}]$$G[VCK] of $$G$$G with exactly one vertex from each partite set and the maximum number of edges or the maximum total edge weight, if $$G$$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 $$\lambda =2$$λ=2, and cannot be approximated within a factor of $$16/17$$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.