General bounds for incremental maximization

Abstract

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $$k\in {\mathbb {N}}$$ k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

Incremental algorithms for the maximum internal spanning tree problem

“…Incremental problems are relevant in multiple real-world applications such as when budget limitations only allow a small expansion of the current local solution at each step. After the initial work by Mettu et al [20], the incremental versions of combinatorial problems have been investigated by multiple researchers [20][21][22][23][24]. In the case of IMIST, the aim is to obtain a tree sequence T 1 , T 2 , .…”

Fractionally Subadditive Maximization under an Incremental Knapsack Constraint

“…In the following, we fix a ground set E, a monotone, M -bounded and fractionally subadditive objective f : 2 E → R ≥0 , and weights w : E → R ≥0 . We present a refined variant of the incremental algorithm introduced in [3]. On a high level, the idea is to consider optimum solutions of increasing sizes, and to add all elements in these optimum solutions one solution at a time.…”

“…In this way, we can guarantee that either the solution we have assembled most recently, or the solution we are currently assembling provides sufficient value to stay competitive. While the algorithm of [3] only scales the capacity, we simultaneously scale capacities and solution values. In addition, we use a more sophisticated order in which we assemble solutions, based on a primal-dual LP formulation.…”

“…Related Work Bernstein et al [3] considered a closely related framework for incremental maximization. Their framework assumes a growing cardinality constraint, which is a special case of our problem in (1) when all elements e ∈ E have unit weight w(e) = 1.…”

Fractionally Subadditive Maximization Under an Incremental Knapsack Constraint

“…To circumvent this issue, Bernstein et al [1] consider accountable functions, i.e., functions f , such that, for every X ⊆ U , there exists e ∈ X with f (X \ {e}) ≥ f (X) − f (X)/|X|. They further show that many natural incremental optimization problems are monotone and accountable such as the following.…”