Combinatorial incremental problems

Abstract

We study the class of Incremental Combinatorial optimization problems, where solutions are evaluated as they are built, as opposed to only measuring the performance of the final solution. Even though many of these problems have been studied, it has' usually been in isolation, so the first objective of this document is to present them under the same framework. We present the incremental analog of several classic combinatorial problems, and present efficient algorithms to find approximate solutions to some of these problems, either improving, or giving the first known approximation guarantees. We present unifying techniques that work for general classes of incremental optimization problems, using fundamental properties of the underlying problem, such as monotonicity or convexity, and relying on algorithms for the non-incremental version of the problem as subroutines. In Chapter 2 we give an e-approximation algorithm for general incremental minimization problems, improving the best approximation guarantee for the incremental version of the shortest path problem. In Chapter 3 we show constant approximation algorithms for several subclasses of incremental maximization problems, including e/2e-1approximation for the maximum weight matching problem, and a e/e+1 approximation for submodular valuations. In Chapter 4 we introduce a discrete-concavity property that allows us to give constant approximation guarantees to several problems, including an asymptotic 0.85-approximation for the incremental maximum flow with unit capacities, and a 0.9-approximation for incremental maximum cardinality matching, incremental maximum stable set in claw free graphs and incremental maximum size common independent set of two matroids.