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Mathematics > Optimization and Control
Title: Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator
(Submitted on 15 Mar 2023 (v1), last revised 25 Apr 2024 (this version, v4))
Abstract: We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lov\'asz-Schrijver SDP operator $\text{LS}_+$. In particular, we focus on a search for relatively small graphs with high $\text{LS}_+$-rank (i.e., the least number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor. This improves upon the previous best result in this direction from 1999, which yielded graphs whose $\text{LS}_+$-rank only grew with the square root of the number of vertices.
Submission history
From: Yu Hin (Gary) Au [view email][v1] Wed, 15 Mar 2023 22:47:55 GMT (94kb,D)
[v2] Wed, 29 Mar 2023 14:54:54 GMT (96kb,D)
[v3] Wed, 3 Jan 2024 00:19:32 GMT (78kb,D)
[v4] Thu, 25 Apr 2024 01:31:46 GMT (79kb,D)
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