# Efficient Approximation for Restricted Biclique Cover Problems

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## Abstract

**:**

## 1. Introduction

#### Motivation

## 2. Overview of New Results

#### 2.1. Model I: A Node belongs to a Small Number of Bicliques

**Problem**

**1**

**.**Given a bipartite graph $G=(L,R,E)$, and parameters ${f}_{L},{f}_{R}$ and k, distinguish between the following two cases:

**Results**

**1.**

#### 2.2. Model II: A One-Side Stochastic Model

**Results**

**2.**

## 3. Model I: $(f,k)$-Biclique Cover

#### 3.1. $(1,k)$-Biclique Cover

**Lemma**

**1.**

**Proof.**

#### 3.2. Results for the $(2,k)$-Biclique Cover

#### 3.2.1. Maximize Edge Coverage with k Bicliques

**Theorem**

**1.**

- Step 1: Kernelization.

**Lemma**

**2.**

**Proof.**

Algorithm 1 ApproxMaxCover($G=(L,R,E),k$) |

Assumes G is a yes-instance for $(2,k)$-biclique cover. — Step 1 — Let ${G}^{\prime}=({L}^{\prime},{R}^{\prime},{E}^{\prime})$ be the graph obtained applying the Fleischner et al. [2] kernelization. Store mapping $\mathtt{kern}$. Let ${V}^{\prime}={L}^{\prime}\cup {R}^{\prime}$ if $|{V}^{\prime}|\ge 2{k}^{2}$ thenreport no $(2,k)$-biclique cover exists for G and quit.end if— Step 2 — Let ${\mathcal{B}}^{\prime}\leftarrow \varnothing $. For all $A\subseteq {V}^{\prime}$ such that $G\left[A\right]={K}_{3,4}$ or $G\left[A\right]={K}_{4,3}$. - Let ${A}_{L}=A\cap {L}^{\prime}$ and ${A}_{R}=A\cap {R}^{\prime}$
- Let ${A}_{L}\leftarrow {\bigcap}_{v\in {A}_{R}}N\left(v\right)$
- Let ${A}_{R}\leftarrow {\bigcap}_{v\in {A}_{L}}N\left(v\right)$
- Let ${A}^{\prime}=({A}_{L},{A}_{R})$ and add biclique ${A}^{\prime}$ to the set ${\mathcal{B}}^{\prime}$
if $|{\mathcal{B}}^{\prime}|\ge k$ thenreport no $(2,k)$-biclique cover exists for G and quit.end if— Step 3 — Let ${\mathcal{C}}^{\prime}$ be the set containing all bicliques of ${G}^{\prime}$ of the form: - All the bicliques $\left(\right\{u\}$,$N\left(u\right))$ for $u\in {L}^{\prime}$ or $\left(N\right(u),\{u\left\}\right)$ for $u\in {R}^{\prime}$
- All the bicliques ($\{u,v\}$, $N\left(u\right)\cap N\left(v\right)$) for $u,v\in {L}^{\prime}$ or $\left(N\right(u)\cap N(v),\{u,v\left\}\right)$ for $u,v\in {R}^{\prime}$
- And all the ${K}_{3,3}$ in ${G}^{\prime}$.
Let $\mathcal{B}=\{\mathtt{kern}\left({B}^{\prime}\right)|{B}^{\prime}\in {\mathcal{B}}^{\prime}\}$. Let $\mathcal{C}=\{\mathtt{kern}\left({C}^{\prime}\right)|{C}^{\prime}\in {\mathcal{C}}^{\prime}\}$. Let $\mathcal{O}=\mathcal{B}$. while $|\mathcal{O}|<k$ doSelect $C\in \mathcal{C}$ s.t. the number of edges covered in G by the union of the bicliques in $\mathcal{O}\cup \left\{C\right\}$ is maximized. Add C to $\mathcal{O}$. end whilereturn$\mathcal{O}$. |

**Lemma**

**3.**

- ${G}^{\prime}$ admits a $(2,k)$-biclique cover.
- $|{L}^{\prime}\cup {R}^{\prime}|\le O\left({k}^{2}\right)$.
- In any $(2,k)$-biclique cover of ${G}^{\prime}$ there are no two nodes on the same side of the graph covered by the exact same set of bicliques.

**Proof.**

- Step 2: Identifying Large Bicliques in the Cover.

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

- Step 3: Small Bicliques

- Step 4: Max Coverage

**Proof of Theorem**

**1.**

#### 3.2.2. Covering All Edges with the Minimum Number of Bicliques

**Theorem**

**2.**

#### Properties of the Kernelized Graph of a $(2,k)$-Biclique Cover Yes-Instance

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

#### Algorithm

**Lemma**

**8.**

**Proof.**

#### 3.3. NP-Complete Result for $f\ge 5$

**Theorem**

**3.**

#### 3.4. Proof of Theorem 3

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

## 4. Model II: One-Side Stochastic Model

#### 4.1. Optimal $(f,\overline{p},k)$-Biclique Cover

**Theorem**

**4.**

#### 4.1.1. Determining Whether Nodes in $S\subseteq L$ Share a Common factor

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Proof.**

#### 4.1.2. Algorithm Based on the Oracle I

Algorithm 2 OptCoverOneSideRandom($G=(L,R,E),f,{p}_{m},k$) |

Mark all edges in E as uncovered. Let $\mathcal{B}\leftarrow \varnothing $ be the set of bicliques in the cover while $\exists u\in L$ adjacent to an uncovered edge do— Step 1 — ${L}_{B}\leftarrow \left\{u\right\}$. — Step 2 — while $\exists S\subseteq L\backslash {L}_{B}$ s.t. $1\le |S|\le f$, $\mathcal{I}(S\cup {L}_{B})=1$ and there is no biclique in $\mathcal{B}$ that covers all nodes in $S\cup {L}_{B}$ do${L}_{B}\leftarrow {L}_{B}\cup S$. end while— Step 3 — for $\forall v\in L$ doif $v\notin {L}_{B}$ and $\mathcal{I}({L}_{B}\cup v)=1$ then${L}_{B}\leftarrow {L}_{B}\cup \left\{u\right\}$. end ifend for— Step 4 — ${R}_{B}={\bigcap}_{u\in {L}_{B}}N\left(u\right)$ $B\leftarrow ({L}_{B},{R}_{B})$ $\mathcal{B}\leftarrow \mathcal{B}\cup B$ Mark all edges in B as covered. end whilereturn $\mathcal{B}$ |

#### Intuition of the Algorithm

#### Correctness of the Algorithm

**Lemma**

**14.**

**Proof.**

**Theorem**

**5.**

**Proof.**

#### Time Complexity

#### 4.2. Approximate Biclique Cover

**Theorem**

**6.**

#### Algorithm

Algorithm 3 ApproxCoverOneSideRandom($G=(L,R,E),f,{p}_{m},{p}_{M},k$) |

Let
$$\overline{c}=max\left(\right)open="("\; close=")">16{p}_{m}^{-1},2{p}_{m}^{-1}{log}_{{(1-{(1-{p}_{M})}^{f})}^{-1}}\left(3\right)$$
Sample a set $S\subseteq R$ of $\overline{c}log(|L|)$ nodes u.a.r. from R (without replacement). Let $\mathcal{C}\leftarrow \varnothing $ for Each subset ${S}^{\prime}\subseteq S$ doLet ${L}^{\prime}\leftarrow {\bigcap}_{v\in {S}^{\prime}}N\left(v\right)$. Let ${S}^{\prime \prime}\leftarrow {\bigcap}_{v\in {L}^{\prime}}N\left(v\right)$. Let B be the set of the edges covered by biclique $({L}^{\prime},{S}^{\prime \prime})$. $\mathcal{C}\leftarrow \mathcal{C}\cup B$. end forApply the greedy max coverage (or set cover) algorithm using $\mathcal{C}$ as the input sets and E as the universe set to cover. |

**Lemma**

**15.**

**Proof.**

**Lemma**

**16.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Chalermsook, P.; Heydrich, S.; Holm, E.; Karrenbauer, A. Nearly Tight Approximability Results for Minimum Biclique Cover and Partition. In Algorithms-ESA 2014; Springer: Berlin, Germany, 2014; pp. 235–246. [Google Scholar]
- Fleischner, H.; Mujuni, E.; Paulusma, D.; Szeider, S. Covering graphs with few complete bipartite subgraphs. Theor. Comput. Sci.
**2009**, 410, 2045–2053. [Google Scholar] [CrossRef][Green Version] - Simon, H.U. On approximate solutions for combinatorial optimization problems. SIAM J. Discret. Math.
**1990**, 3, 294–310. [Google Scholar] [CrossRef] - Gruber, H.; Holzer, M. Inapproximability of nondeterministic state and transition complexity assuming P ≠ NP. In Developments in Language Theory; Springer: Berlin, Germany, 2007; pp. 205–216. [Google Scholar]
- Orlin, J. Contentment in graph theory: Covering graphs with cliques. In Indagationes Mathematicae (Proceedings); Elsevier: NewYork, NY, USA, 1977; Volume 80, pp. 406–424. [Google Scholar]
- Jukna, S.; Kulikov, A.S. On covering graphs by complete bipartite subgraphs. Discret. Math.
**2009**, 309, 3399–3403. [Google Scholar] [CrossRef][Green Version] - Nor, I.; Hermelin, D.; Charlat, S.; Engelstadter, J.; Reuter, M.; Duron, O.; Sagot, M.F. Mod/Resc parsimony inference: Theory and application. Inf. Comput.
**2012**, 213, 23–32. [Google Scholar] [CrossRef] - Nor, I.; Engelstädter, J.; Duron, O.; Reuter, M.; Sagot, M.F.; Charlat, S. On the genetic architecture of cytoplasmic incompatibility: Inference from phenotypic data. Am. Nat.
**2013**, 182, E15–E24. [Google Scholar] [CrossRef] [PubMed] - Nau, D.S.; Markowsky, G.; Woodbury, M.A.; Amos, D.B. A mathematical analysis of human leukocyte antigen serology. Math. Biosci.
**1978**, 40, 243–270. [Google Scholar] [CrossRef] - Miettinen, P.; Mielikainen, T.; Gionis, A.; Das, G.; Mannila, H. The discrete basis problem. IEEE Trans. Knowl. Data Eng.
**2008**, 20, 1348–1362. [Google Scholar] [CrossRef] - Mishra, N.; Ron, D.; Swaminathan, R. Learning Theory and Kernel Machines. In Proceedings of the 16th Annual Conference on Learning Theory and 7th Kernel Workshop COLT/Kernel 2003, Washington, DC, USA, 24–27 August 2003; pp. 448–462. [Google Scholar]
- Hirsch, M.; Meijer, H.; Rappaport, D. Biclique edge cover graphs and confluent drawings. In Graph Drawing; Springer: Berlin, Germany, 2006; pp. 405–416. [Google Scholar]
- Müller, H. On edge perfectness and classes of bipartite graphs. Discret. Math.
**1996**, 149, 159–187. [Google Scholar] [CrossRef] - Amilhastre, J.; Vilarem, M.C.; Janssen, P. Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs. Discret. Appl. Math.
**1998**, 86, 125–144. [Google Scholar] [CrossRef] - Dawande, M.; Keskinocak, P.; Swaminathan, J.M.; Tayur, S. On bipartite and multipartite clique problems. J. Algorithms
**2001**, 41, 388–403. [Google Scholar] [CrossRef] - Javadi, R.; Maleki, Z.; Omoomi, B. Local Clique Covering of Graphs. arXiv, 2012; arXiv:1210.6965. [Google Scholar]
- Arora, S.; Ge, R.; Sachdeva, S.; Schoenebeck, G. Finding overlapping communities in social networks: Toward a rigorous approach. In Proceedings of the 13th ACM Conference on Electronic Commerce, Valencia, Spain, 4–8 June 2012; pp. 37–54. [Google Scholar]
- Cheng, Y.; Church, G.M. Biclustering of Expression Data; ISMB: Leesburg, VA, USA, 2000; Volume 8, pp. 93–103. [Google Scholar]
- Král’, D.; Kratochvíl, J.; Tuza, Z.; Woeginger, G. Complexity of coloring graphs without forbidden induced subgraphs. In Graph-Theoretic Concepts in Computer Science; Springer: Berlin, Germany, 2001; pp. 254–262. [Google Scholar]
- Fellows, M.R.; Kratochvíl, J.; Middendorf, M.; Pfeiffer, F. The complexity of induced minors and related problems. Algorithmica
**1995**, 13, 266–282. [Google Scholar] [CrossRef]

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Epasto, A.; Upfal, E.
Efficient Approximation for Restricted Biclique Cover Problems. *Algorithms* **2018**, *11*, 84.
https://doi.org/10.3390/a11060084

**AMA Style**

Epasto A, Upfal E.
Efficient Approximation for Restricted Biclique Cover Problems. *Algorithms*. 2018; 11(6):84.
https://doi.org/10.3390/a11060084

**Chicago/Turabian Style**

Epasto, Alessandro, and Eli Upfal.
2018. "Efficient Approximation for Restricted Biclique Cover Problems" *Algorithms* 11, no. 6: 84.
https://doi.org/10.3390/a11060084