Simpler Proofs by Symbolic Perturbation

TL;DR

This paper demonstrates that many algorithms for weighted combinatorial optimization can be made robust against special cases by using symbolic perturbation, simplifying their analysis and ensuring consistent performance.

Contribution

It introduces the concept of null cases and shows that symbolic perturbation guarantees robustness for a broad class of algorithms.

Findings

Symbolic perturbation ensures no surprises in null cases.

Robustness can be achieved by breaking ties correctly.

Any algorithm admits a symbolic perturbation tie-breaking policy.

Abstract

In analyses of algorithms, a substantial amount of effort has often to be spent on the discussion of special cases. For example, when the analysis considers the cases X<Y and X>Y separately, one might have to be especially careful about what happens when X=Y. On the other hand, experience tells us that when a yet unregarded special case of this kind is discovered, one nearly always finds a way to handle it. This is typically done by modifying the analysis and/or the algorithm very slightly. In this article we substantiate this observation theoretically. We concentrate on deterministic algorithms for weighted combinatorial optimization problems. A problem instance of this kind is defined by its structure and a vector of weights. The concept of a null case is introduced as set of problem instances whose weight vectors constitute a nowhere open set (or null set) in the space of all possible weight configurations. An algorithm is called robust if any null case can be disregarded in the analysis of both its solution quality and resource requirements. We show that achieving robustness is only a matter of breaking ties the right way. More specifically, we show that the concept of symbolic perturbation known from the area of geometric algorithms guarantees that no surprises will happen in null cases. We argue that for a huge class of combinatorial optimization algorithms it is easy to verify that they implicitly use symbolic perturbation for breaking ties and thus can be analyzed under the assumption that some arbitrary null case never occurs. Finally, we prove that there exists a symbolic perturbation tie breaking policy for any algorithm.