### Abstract

We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U, V, E) and a parameter β, the goal is to find a subset V′ ⊆ V containing β fraction of the vertices of V which minimizes the size of N(V), the neighborhood of V′. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion. In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants τ, γ > 0 there is no algorithm which, given a constant β > 0 and a bipartite graph G(U, V, E), runs in polynomial time and decides whether • (YES case) There is a subset S^{∗} ⊆ V s.t. S^{∗}| ≥ β |V| satisfying | N(S^{∗})| ≤ γ |U|, or • (NO case) Any subset S ⊆ V s.t. |S| ≥ τβ|V| satisfies | N(S)| ≥ (1 - γ|U|, unless NP ⊆ ∩_{ϵ}oDTIME (2^{nϵ}) i.e. NP has subexponential time algorithms. We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of Raghavendra and Steurer [23].

Original language | English (US) |
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Title of host publication | 24th Annual European Symposium on Algorithms, ESA 2016 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 57 |

ISBN (Electronic) | 9783959770156 |

DOIs | |

State | Published - Aug 1 2016 |

Event | 24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark Duration: Aug 22 2016 → Aug 24 2016 |

### Other

Other | 24th Annual European Symposium on Algorithms, ESA 2016 |
---|---|

Country | Denmark |

City | Aarhus |

Period | 8/22/16 → 8/24/16 |

### Fingerprint

### Keywords

- Bipartite expansion
- Inapproximability
- PCP
- Submodular minimization

### ASJC Scopus subject areas

- Software

### Cite this

*24th Annual European Symposium on Algorithms, ESA 2016*(Vol. 57). [55] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2016.55

**Hardness of bipartite expansion.** / Khot, Subhash; Saket, Rishi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*24th Annual European Symposium on Algorithms, ESA 2016.*vol. 57, 55, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 24th Annual European Symposium on Algorithms, ESA 2016, Aarhus, Denmark, 8/22/16. https://doi.org/10.4230/LIPIcs.ESA.2016.55

}

TY - GEN

T1 - Hardness of bipartite expansion

AU - Khot, Subhash

AU - Saket, Rishi

PY - 2016/8/1

Y1 - 2016/8/1

N2 - We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U, V, E) and a parameter β, the goal is to find a subset V′ ⊆ V containing β fraction of the vertices of V which minimizes the size of N(V), the neighborhood of V′. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion. In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants τ, γ > 0 there is no algorithm which, given a constant β > 0 and a bipartite graph G(U, V, E), runs in polynomial time and decides whether • (YES case) There is a subset S∗ ⊆ V s.t. S∗| ≥ β |V| satisfying | N(S∗)| ≤ γ |U|, or • (NO case) Any subset S ⊆ V s.t. |S| ≥ τβ|V| satisfies | N(S)| ≥ (1 - γ|U|, unless NP ⊆ ∩ϵoDTIME (2nϵ) i.e. NP has subexponential time algorithms. We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of Raghavendra and Steurer [23].

AB - We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U, V, E) and a parameter β, the goal is to find a subset V′ ⊆ V containing β fraction of the vertices of V which minimizes the size of N(V), the neighborhood of V′. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion. In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants τ, γ > 0 there is no algorithm which, given a constant β > 0 and a bipartite graph G(U, V, E), runs in polynomial time and decides whether • (YES case) There is a subset S∗ ⊆ V s.t. S∗| ≥ β |V| satisfying | N(S∗)| ≤ γ |U|, or • (NO case) Any subset S ⊆ V s.t. |S| ≥ τβ|V| satisfies | N(S)| ≥ (1 - γ|U|, unless NP ⊆ ∩ϵoDTIME (2nϵ) i.e. NP has subexponential time algorithms. We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of Raghavendra and Steurer [23].

KW - Bipartite expansion

KW - Inapproximability

KW - PCP

KW - Submodular minimization

UR - http://www.scopus.com/inward/record.url?scp=85012960878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012960878&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2016.55

DO - 10.4230/LIPIcs.ESA.2016.55

M3 - Conference contribution

VL - 57

BT - 24th Annual European Symposium on Algorithms, ESA 2016

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -