Incremental Network Optimization: Theory and Algorithms

Abstract

In an incremental optimization problem, we are given a feasible solution x0 of an optimization problem P, and we want to make an incremental change in x0 that will result in the greatest improvement in the objective function. In this paper, we study the incremental optimization versions of six well-known network problems. We present a strongly polynomial algorithm for the incremental minimum spanning tree problem. We show that the incremental minimum cost flow problem and the incremental maximum flow problem can be solved in polynomial time using Lagrangian relaxation. We consider two versions of the incremental minimum shortest path problem, where increments are measured via arc inclusions and arc exclusions. We present a strongly polynomial time solution for the arc inclusion version and show that the arc exclusion version is NP-complete. We show that the incremental minimum cut problem is NP-complete and that the incremental minimum assignment problem reduces to the minimum exact matching problem, for which a randomized polynomial time algorithm is known.

Robust recoverable 0–1 optimization problems under polyhedral uncertainty

“…It is worth noting that all the neighborhoods, based on the element inclusion, the element exclusion and the symmetric difference, considered in [12] are equivalent if |I(x x x)| = m, m ∈ [n], for each x x x ∈ X . We will call such a problem P, an equal cardinality problem.…”

“…In this paper we will also consider the following evaluation problem: The computational complexity of the incremental, recoverable, robust recoverable and adversarial problems is not less than the complexity of the generic problem P. So, all these problems are NP-hard if P is already NP-hard. However, even the incremental version of P can be much harder than P. It has been shown in [12], then the incremental shortest path problem (X is the set of characteristic vectors of simple paths in a given graph) for the element exclusion neighborhood is NP-hard and hard to approximate (interestingly, the incremental shortest path problem with the element inclusion neighborhood is polynomially solvable [12]). The incremental minimum assignment problem (X is the set of characteristic vectors of perfect matchings in a bipartite graph) is equivalent to the minimum exact matching problem, for which no polynomial time algorithm is known [12].…”

Incremental computations over strongly distributed databases

“…In addition, assume that the query language allows us to ask optimization questions. In such cases, our generalized propagation technique may be directly extended to the case of incremental optimization as considered in . Using our method, the final optimal result is computed from the local (not necessarily optimal) solutions.…”

The recoverable robust spanning tree problem with interval costs is polynomially solvable

“…It is easy to see that this problem is the inner one in RR ST, where X is fixed and U contains only one scenario. The incremental spanning tree problem can be solved in polynomial time by applying the Lagrangian relaxation technique [12]. In [2] a polynomial algorithm for a more general recoverable robust matroid basis problem (RR MB, for short) with scenario set U m was proposed, provided that the recovery parameter k is constant and, in consequence, for RR ST (a spanning tree is a graphic matroid).…”