#### Theory Seminar

# New Approximation Bounds for Small-Set Vertex Expansion

Suprovat GhoshalNorthwestern / TTIC

WHERE:

3725 Beyster Building

WHEN:

Friday, January 12, 2024 @ 2:00 pm - 3:00 pm

This event is free and open to the publicAdd to Google Calendar

This event is free and open to the publicAdd to Google Calendar

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(Passcode: 430018)

The vertex expansion of the graph is a fundamental graph parameter. Given a graph and a parameter , its -Small-Set Vertex Expansion (SSVE) is defined as

where is the vertex boundary of a set . 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 StrongUniqueGames Problem (Ghoshal-Louis’21). We give a randomized algorithm running in time , which outputs a set of size , having vertex expansion at most

where is the largest vertex degree of the graph, and is the optimal -SSVE. The previous best-known guarantees for this were the bi-criteria bounds of and due to Louis-Makarychev [TOC’16].

Our algorithm uses the basic SDP relaxation of the problem augmented with rounds of the Lasserre/SoS hierarchy. Our rounding algorithm is a combination of the rounding algorithms of Raghavendra-Tan’12 and Austrin-Benabbas-Georgiou’13. A key component of our analysis is novel Gaussian rounding lemma for hyperedges, which might be of independent interest.

This is joint work with Anand Louis.