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Improved Kernelization and Fixed-parameter Algorithms for Bicluster Editing

Manuel Lafond

TL;DR

This work shows that one can obtain a kernel with 4.5k vertices, an improvement over the previously known quadratic kernel and proposes an algorithm that runs in time O∗(2.581k), which has the advantage of being conceptually simple and should be easy to implement.

Abstract

Given a bipartite graph $G$, the \textsc{Bicluster Editing} problem asks for the minimum number of edges to insert or delete in $G$ so that every connected component is a bicluster, i.e. a complete bipartite graph. This has several applications, including in bioinformatics and social network analysis. In this work, we study the parameterized complexity under the natural parameter $k$, which is the number of allowed modified edges. We first show that one can obtain a kernel with $4.5k$ vertices, an improvement over the previously known quadratic kernel. We then propose an algorithm that runs in time $O^*(2.581^k)$. Our algorithm has the advantage of being conceptually simple and should be easy to implement.

Improved Kernelization and Fixed-parameter Algorithms for Bicluster Editing

TL;DR

This work shows that one can obtain a kernel with 4.5k vertices, an improvement over the previously known quadratic kernel and proposes an algorithm that runs in time O∗(2.581k), which has the advantage of being conceptually simple and should be easy to implement.

Abstract

Given a bipartite graph , the \textsc{Bicluster Editing} problem asks for the minimum number of edges to insert or delete in so that every connected component is a bicluster, i.e. a complete bipartite graph. This has several applications, including in bioinformatics and social network analysis. In this work, we study the parameterized complexity under the natural parameter , which is the number of allowed modified edges. We first show that one can obtain a kernel with vertices, an improvement over the previously known quadratic kernel. We then propose an algorithm that runs in time . Our algorithm has the advantage of being conceptually simple and should be easy to implement.

Paper Structure

This paper contains 4 sections, 16 theorems, 7 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary notions.
  3. A $4.5k$ kernel
  4. An $O^*(2.581^k)$ branching algorithm

Key Result

lemma thmcounterlemma

There exists an optimal biclustering $\mathcal{B}$ of $G$ such that, for every twin class $R$ of $G$, the vertices of $R$ are all in the same bicluster of $\mathcal{B}$.

Figures (6)

  • Figure 1: $R$ is a twin class, $S = N(R)$, $T$ are the sisters of $R$ (there is only one) and $W = N(S) \setminus (T \cup R)$. Note how the two bottom vertices of $W$ are not sisters of $R$, because they are twins.
  • Figure 2: Left: an illustration of $\mathcal{B}$ and relevant subsets of vertices. The dashed shape indicates $B$, the bicluster that contains $R'$. The fat edge represents deletions in $\mathcal{B}$ not in $\mathcal{B}'$, and the fat dotted edge insertions in $\mathcal{B}$ not in $\mathcal{B}'$. Right: an illustration of $\mathcal{B}'$. The upper dashed shape indicates the new bicluster to introduce, and the lower dashed shape is what remains in $B$ after modification. Fat edges are the worst-case deletions in $\mathcal{B}'$ not in $\mathcal{B}$, the dotted edge represents insertions in $\mathcal{B}'$ not in $\mathcal{B}$.
  • Figure 3: The bicluster introduced in $\mathcal{B}'$ (dashed shape). Fat edges are deletions in $\mathcal{B}'$ not in $\mathcal{B}$, light edges represents the fact that each element of each $T_i$ has at most one neighbor in $U_i$, and they are all distinct. Not shown: insertions between $W_1$ and $S \setminus S_1$.
  • Figure 4: An illustration of the $f$ map. Here, $x \in X$ and $I_f(x) = \{u, x, v\}$. No vertex of $R_i$ other than $u$ maps to $x$, and no vertex of $R_j$ other than $v$ maps to $x$. Dotted lines indicate modified edges incident to $x$. Each of $u, x,$ and $v$ charges $1/2$ to each dotted edge.
  • Figure 5: A graph of size $4.5k$, with $k = 2$, on which none of the reduction rules apply.
  • ...and 1 more figures

Theorems & Definitions (41)

  • proof
  • lemma thmcounterlemma: guo2008improved
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof
  • Claim 1
  • ...and 31 more