Abstract
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 \(V=V_{1}\bigcup \cdots \bigcup V_{n}\) and each partite set \(V_{i}\) containing a constant number \(\lambda \) 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 \(\lambda =2\), 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.
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Notes
The Jaccard similarity of two unweighted graph \(G_{1}=(V_{1},E_{1})\) and \(G_{2}=(V_{2},E_{2})\) is \(\frac{|E_{1}\bigcap E_{2}|}{|E_{1}\bigcup E_{2}|}\). If \(G_{1}\) and \(G_{2}\) are weighted complete graphs (missing edges can be viewed as edges with \(0\) weight), their Jaccard similarity is \(\frac{\sum _{1\le i\le |E_{1}\cup E_{2}|}\min \{w_{i}, w'_{i}\}}{\sum _{1\le i\le |E_{1}\cup E_{2}|}\max \{w_{i}, w'_{i}\}}.\)
Given a set of sets, Jaccard median is the set (which might be a new set, and not from the given sets) with minimum total Jaccard distances to the given sets. Here, each labeled graph can be viewed as a set of edges.
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Acknowledgments
This research was partially supported by NSF through Grants IIS-1115220, IIS-1422591, and CCF-1422324.
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Ding, H., Stojkovic, B., Chen, Z. et al. Chromatic kernel and its applications. J Comb Optim 31, 1298–1315 (2016). https://doi.org/10.1007/s10878-014-9824-z
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DOI: https://doi.org/10.1007/s10878-014-9824-z