Bilu-Linial stability, certified algorithms and the Independent Set problem

TL;DR

This paper investigates the Bilu-Linial stability concept for the Maximum Independent Set problem, providing algorithms and bounds for stable instances across various graph classes and exploring certified algorithms that guarantee optimality under perturbations.

Contribution

The paper introduces new algorithms and bounds for stable MIS instances on multiple graph classes and initiates the study of certified algorithms with provable guarantees.

Findings

Efficient algorithms for stable MIS on graphs with bounded degree, colorability, and planarity.

Lower bounds showing hardness for certain stable instances assuming the planted clique conjecture.

Development of certified algorithms that produce solutions optimal under perturbations.

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 $\gamma\geq 1$. 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 $\left(\frac{\Delta + 1}{3} + \varepsilon\right)$-certified algorithm for MIS on graphs of maximum degree $\Delta$ where all weights are equal to 1.