Bilu-Linial stability, certified algorithms and the Independent Set problem
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We study the notion of Bilu-Linial stability in the context of Independent Set. A weighted instance G=(V,E,w) of Independent Set is γ-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ≥ 1. In this work, we use the standard LP as well as the Sherali-Adams hierarchy to design algorithms for (Δ-1)-stable instances on graphs of maximum degree Δ, for (k-1)-stable instances on k-colorable graphs and for (1+ε)-stable instances on planar graphs. We also show that the integrality gap of relaxations of several maximization problems reduces dramatically on stable instances. For general graphs we give an algorithm for (ε n)-stable instances (for fixed ε>0), and on the negative side we show that there are no efficient algorithms for O(n^1/2-ε)-stable instances assuming the planted clique conjecture. As a side note, we exploit the connection between Vertex Cover and Node Multiway Cut and give the first results about stable instances of Node Multiway Cut. Moreover, we initiate the study of certified algorithms for Independent Set. The class of γ-certified algorithms is a class of γ-approximation algorithms introduced by Makarychev and Makarychev (2018) whose returned solution is optimal for a perturbation of the original instance. Using results of Makarychev and Makarychev (2018) as well as combinatorial techniques, we obtain Δ-certified algorithms for Independent Set on graphs of maximum degree Δ and (1+ε)-certified algorithms on planar graphs. Finally, we prove that an algorithm of Berman and Fürer (1994) is a (Δ+1/3+ε)-certified algorithm on graphs of maximum degree Δ where all weights are equal to 1.
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