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• W2895978147 abstract "We study the Maximum Independent Set (MIS) problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of MIS is $gamma$-stable if it has a unique optimal solution that remains the unique optimum under multiplicative perturbations of the weights by a factor of at most $gammageq 1$. The goal then is to efficiently recover the unique optimal solution. In this work, we solve stable instances of MIS on several graphs classes: we solve $widetilde{O}(Delta/sqrt{log Delta})$-stable instances on graphs of maximum degree $Delta$, $(k - 1)$-stable instances on $k$-colorable graphs and $(1 + varepsilon)$-stable instances on planar graphs. For general graphs, we present a strong lower bound showing that there are no efficient algorithms for $O(n^{frac{1}{2} - varepsilon})$-stable instances of MIS, assuming the planted clique conjecture. We also give an algorithm for $(varepsilon n)$-stable instances. As a by-product of our techniques, we give algorithms and lower bounds for stable instances of Node Multiway Cut. Furthermore, we prove a general result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms, a notion recently introduced by Makarychev and Makarychev (2018), which is a class of $gamma$-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance. We obtain $Delta$-certified algorithms for MIS on graphs of maximum degree $Delta$, and $(1+varepsilon)$-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Furer (1994) and prove that it is a $left(frac{Delta + 1}{3} + varepsilonright)$-certified algorithm for MIS on graphs of maximum degree $Delta$ where all weights are equal to 1." @default.
• W2895978147 created "2018-10-26" @default.
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• W2895978147 date "2018-10-19" @default.
• W2895978147 modified "2023-09-27" @default.
• W2895978147 title "Bilu-Linial stability, certified algorithms and the Independent Set problem" @default.
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