New Approximation Bounds for Small-Set Vertex Expansion
ACM-SIAM Symposium on Discrete Algorithms(2023)
摘要
The vertex expansion of the graph is a fundamental graph parameter. Given a
graph $G=(V,E)$ and a parameter $\delta \in (0,1/2]$, its $\delta$-Small-Set
Vertex Expansion (SSVE) is defined as \[ \min_{S : |S| = \delta |V|}
\frac{|{\partial^V(S)}|}{ \min \{ |S|, |S^c| \} } \] where $\partial^V(S)$ is
the vertex boundary of a set $S$. The SSVE~problem, in addition to being of
independent interest as a natural graph partitioning problem, is also of
interest due to its connections to the Strong Unique Games problem. We give a
randomized algorithm running in time $n^{{\sf poly}(1/\delta)}$, which outputs
a set $S$ of size $\Theta(\delta n)$, having vertex expansion at most \[
\max\left(O(\sqrt{\phi^* \log d \log (1/\delta)}) ,
\tilde{O}(d\log^2(1/\delta)) \cdot \phi^* \right), \] where $d$ is the largest
vertex degree of the graph, and $\phi^*$ is the optimal $\delta$-SSVE. The
previous best-known guarantees for this were the bi-criteria bounds of
$\tilde{O}(1/\delta)\sqrt{\phi^* \log d}$ and $\tilde{O}(1/\delta)\phi^*
\sqrt{\log n}$ due to Louis-Makarychev [TOC'16].
Our algorithm uses the basic SDP relaxation of the problem augmented with
${\rm poly}(1/\delta)$ rounds of the Lasserre/SoS hierarchy. Our rounding
algorithm is a combination of the rounding algorithms of Raghavendra-Tan
[SODA'12] and Austrin-Benabbas-Georgiou [SODA'13]. A key component of our
analysis is novel Gaussian rounding lemma for hyperedges which might be of
independent interest.
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