  The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly
states that it is NP-hard to distinguish between a graph with a small subset of
vertices whose edge expansion is almost zero and one in which all small subsets
of vertices have expansion almost one. In this work, we prove inapproximability
results for the following graph problems based on this hypothesis:
  - Maximum Edge Biclique (MEB): given a bipartite graph $G$, find a complete
bipartite subgraph of $G$ with maximum number of edges.
  - Maximum Balanced Biclique (MBB): given a bipartite graph $G$, find a
balanced complete bipartite subgraph of $G$ with maximum number of vertices.
  - Minimum $k$-Cut: given a weighted graph $G$, find a set of edges with
minimum total weight whose removal partitions $G$ into $k$ connected
components.
  - Densest At-Least-$k$-Subgraph (DAL$k$S): given a weighted graph $G$, find a
set $S$ of at least $k$ vertices such that the induced subgraph on $S$ has
maximum density (the ratio between the total weight of edges and the number of
vertices).
  We show that, assuming SSEH and NP $\nsubseteq$ BPP, no polynomial time
algorithm gives $n^{1 - \varepsilon}$-approximation for MEB or MBB for every
constant $\varepsilon > 0$. Moreover, assuming SSEH, we show that it is NP-hard
to approximate Minimum $k$-Cut and DAL$k$S to within $(2 - \varepsilon)$ factor
of the optimum for every constant $\varepsilon > 0$.
  The ratios in our results are essentially tight since trivial algorithms give
$n$-approximation to both MEB and MBB and efficient $2$-approximation
algorithms are known for Minimum $k$-Cut [SV95] and DAL$k$S [And07, KS09].
  Our first result is proved by combining a technique developed by Raghavendra
et al. [RST12] to avoid locality of gadget reductions with a generalization of
Bansal and Khot's long code test [BK09] whereas our second result is shown via
elementary reductions.

Computational Complexity