We introduce a notion called entropic independence that is an entropic analog of spectral notions of highdimensional expansion. Informally, entropic independence of a background distribution $\mu$ on $k$sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponentialsized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified logSobolev inequalities, for downup random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified logSobolev inequalities, which guarantee entropy contraction not for allmore »
Improving the Smoothed Complexity of FLIP for Max Cut Problems
Finding locally optimal solutions for MAXCUT and MAX k CUT are wellknown PLScomplete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worstcase instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the runtime of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for maxcut in arbitrary graphs is quasipolynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for maxcut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edgeweight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their runtime bound is far from optimal. In this work, we make substantial progress toward improving the runtime bound. We prove that the smoothed complexity of FLIP for maxcut in complete graphs is O (Φ n 7.83 ). Our results are based on a carefully chosen matrix whose rank captures the more »
 Award ID(s):
 1814613
 Publication Date:
 NSFPAR ID:
 10293526
 Journal Name:
 ACM Transactions on Algorithms
 Volume:
 17
 Issue:
 3
 Page Range or eLocationID:
 1 to 38
 ISSN:
 15496325
 Sponsoring Org:
 National Science Foundation
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