Bicluster Editing with Overlaps: A Vertex Splitting Approach

TL;DR

This work introduces BCEVS and BCEOVS to extend BiCluster Editing by allowing vertex splitting to create overlapping biclusters, with two variants depending on whether splits are allowed on both sides or only on one side. It establishes a comprehensive complexity landscape: BCEVS and BCEOVS are NP-complete and APX-hard even for bipartite planar graphs with max degree three, with ETH-based lower bounds; BCEOVS is fixed-parameter tractable with a polynomial kernel and both problems are solvable in polynomial time on trees. The paper also provides a tree-based DP algorithm, a careful parameterized analysis, and reductions from MAX-3-SAT(4) to prove approximation hardness. Together, these results map the computational feasibility of overlapping biclustering under edge edits and vertex-splitting operations and highlight fertile directions for future algorithmic improvements and parameterized studies.

Abstract

The BiCluster Editing problem aims at editing a given bipartite graph into a disjoint union of bicliques via a minimum number of edge deletion or addition operations. As a graph-based model for data clustering, the problem aims at a partition of the input dataset, which cannot always obtain meaningful clusters when some data elements are expected to belong to more than one cluster each. To address this limitation, we introduce the Bicluster Editing with Vertex Splitting problem (BCEVS) which consists of finding a minimum sequence of edge editions and vertex splittings such that the resulting graph is a disjoint union of bicliques. The vertex splitting operation consists of replacing a vertex $v$ with two vertices whose union of neighborhoods is the neighborhood of $v$. We also introduce the problem of Bicluster Editing with One-Sided Vertex Splitting (BCEOVS) where we restrict the splitting operations to the only one set of the two sets forming the bipartition. We prove that the two problems are NP-complete even when restricted to bipartite planar graphs of maximum degree three. Moreover, assuming the {\sc Exponential Time Hypothesis} holds, there is no $2^{o(n)}n^{O(1)}$-time (resp. $2^{o(\sqrt{n})}n^{O(1)}$-time) algorithm for BCEVS and BCEOVS on bipartite (resp. planar) graphs with maximum degree three, where $n$ is the number of vertices of the graph. Furthermore we prove both problems are APX-hard and solvable in polynomial time on trees. On the other hand, we prove that BCEOVS is fixed-parameter tractable with respect to solution size by showing that it admits a polynomial size kernel.

Figures (10)

• Figure 1: Example of a bipartite graph such that $bcevs(G) < bceovs(G,A)$. Here we have $bcevs(G) = 2$ and $bceovs(G,A) = 3$ because in the first case we can split two vertices, one from each side, and in the second case we can split one vertex and delete two edges.
• Figure 2: Example in the cycle $C_{12}$ where two edge deletions occur at indices that are not equal modulo 3. As there are 3 independent geodesics of length 4 (in green), we need at least 5 operations in this case.
• Figure 3: The graph constructed by Construction \ref{['construction:reduction']} for the 3-CNF $(a \vee b \vee c) \land (a \vee b \vee \overline{c})$.
• Figure 4: Example of a clause $c$ where the three green edges incident to $c$ are not deleted and the three green edges incident to $v_c$ are not deleted. The edges in gray should be added or the vertex $v_c$ should be split.
• Figure 5: Example of a tree with a cut vertex $y$ connected to only one vertex $z$ in $Y$ and such that $X$ is a star. An optimal solution consists here in deleting the edge $yz$.

Theorems & Definitions (37)

• lemma 1
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