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 = V 1 ⋃ ⋯ ⋃ V n and each partite set $$V_{i}$$ V i containing a constant number $$\lambda $$ λ of vertices, compute a subgraph $$G[V_{CK}]$$ G [ V C K ] 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.