Multiplying Pessimistic Estimators: Deterministic Approximation of Max TSP and Maximum Triangle Packing

We give a generalization of the method of pessimistic estimators [14], in which we compose estimators by multiplying them. We give conditions on the pessimistic estimators of two expectations, under which the product of the pessimistic estimators is a pessimistic estimator of the product of the two expectations. This approach can be useful when derandomizing algorithms for which one needs to bound a certain probability, which can be expressed as an intersection of multiple events; using our method, one can define pessimistic estimators for the probabilities of the individual events, and then multiply them to obtain a pessimistic estimator for the probability of the intersection of the events. We apply this method to give derandomizations of all known approximation algorithms for the maximum traveling salesman problem and the maximum triangle packing problem: we define simple pessimistic estimators based on the analysis of known randomized algorithms and show that we can multiply them to obtain pessimistic estimators for the expected weight of the solution. This gives deterministic algorithms with better approximation guarantees than what was previously known.