A Modern View on Stability of Approximation

Abstract

In order to attack hard optimization problems that do not admit any polynomial-time approximation scheme (PTAS) or α-approximation algorithm for a reasonable constant α (or even with a worse approximability), Hromkovi£ et al. [12,31] introduced the notion of Stability of Approximation. The main idea of the stability concept is to try to split the set of all input instances into (potentially innitely many) classes with respect to the achievable approximation ratio. The crucial point in applying this concept is to nd parameters that well capture the diculty of problem instances of the particular hard problem. The concept of stability of approximation turns out to be ubiquitous in the research of approximation algorithms and is applicable beyond approximation. In the literature, however, the relation to stability of approximation has stayed implicit. The purpose of this survey is to take a broader view and to explicitly analyze algorithmic problems and their algorithms with respect to stability of approximation.I would like to express my belief that, in order to make essential progress in algorithmics, one has to move from measuring the problem hardness in a worst-case manner to classifying the hardness of the instances of the investigated algorithmic problems.

On the Approximability of the Single Allocation p-Hub Center Problem with Parameterized Triangle Inequality

“…Our study uses the well-known concept of stability of approximation for hard optimization problems [9,11,22,23,25]. The idea of this concept is similar to that of the stability of numerical algorithms.…”

From Symmetry to Asymmetry: Generalizing TSP Approximations by Parametrization

“…Probably the most extensively studied stability measure for approximation of (A)TSP is the β-triangle inequality, also called parameterized triangle inequality, which refers to the requirement c(u, v) ≤ β(c(u, w) + c(w, v)) for all u, v, w ∈ V with u = v = w. For ATSP with β-triangle inequality, the 1 2(1−β) -approximation derived by Kowalik and Mucha [23] for β ∈ ( 1 2 , 1) improves upon a series on previous results [8,4,31] and is also known to be tight with respect to using the cycle cover relaxation as lower bound. For TSP, a recent survey of Klasing and Mömke [22] gives a summary of the known results about TSP with β-triangle inequality.…”

On the Hardness and Approximation of the Densest $K$-Subgraph Problem in Parameterized Metric Graphs

“…The design of approximation algorithms for the ∆-WDkS problem is related to the concept of stability of approximation for hard optimization problems [11], [15], [30], [31], [34]. It is similar to that of the stability of numerical algorithms.…”