### Certified Algorithms: Worst-Case Analysis and Beyond

Konstantin Makarychev, Yury Makarychev, Michael Wagner
2020 Innovations in Theoretical Computer Science
In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a γ-certified algorithm is also a γ-approximation algorithm -it finds a γ-approximation no matter what the input is. Second, it exactly solves γ-perturbation-resilient instances (γ-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both
more » ... mization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient 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, k-medians and k-means. We also present some negative results. Note that if γ = 1, then I = I. Loosely speaking, the closer γ to 1 is, the closer I to I is. All problems we consider are scale invariant, so it will not matter whether we use formula (1) or (2). It will be convenient to use (1) for combinatorial optimization problems and (2) for clustering problems. The central definition of this paper is that of a certified algorithm. Definition 2. A γ-certified solution for instance I is a pair (I , s * ), where I is a γperturbation of I and s * is an optimal solution for I . A γ-certified algorithm (or a γ-certified approximation algorithm) is an algorithm that finds a γ-certified solution. 1