In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worstcase and beyondworstcase performance guarantees. First, a γcertified algorithm is also a γapproximation algorithm  it finds a γapproximation no matter what the input is. Second, it exactly solves γperturbationresilient instances (γperturbationresilient instances model reallife instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbationresilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, kmedians and kmeans. We also present some negative results.
BiluLinial Stability, Certified Algorithms and the Independent Set Problem
We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γstable 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. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))stable instances on graphs of maximum degree ∆, (k − 1)stable instances on kcolorable graphs and (1 + ε)stable instances on planar graphs (for any fixed ε > 0), using both combinatorial techniques as well as LPs and the SheraliAdams hierarchy.
For general graphs, we give an algorithm for (εn)stable instances, for any fixed ε > 0, and lower bounds based on the planted clique conjecture. As a byproduct of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality more »
 Publication Date:
 NSFPAR ID:
 10112618
 Journal Name:
 27th Annual European Symposium on Algorithms (ESA 2019)
 Volume:
 27
 Page Range or eLocationID:
 25:1–25:16
 Sponsoring Org:
 National Science Foundation
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