A typical algorithm for a combinatorial optimization problem is given a static set of input, and ... more A typical algorithm for a combinatorial optimization problem is given a static set of input, and finds a single solution. This kind of algorithm cannot be used, however, in situations where the input and solutions change over time due to evolving constraints. This dissertation explores incremental problems, one class of optimization problems that deals with this type of situation. An incremental algorithm is given a sequence of input, and finds a sequence of solutions that build incrementally while adapting to the changes in the input.
An algorithm for a typical combinatorial optimization problem is given static input, and finds a ... more An algorithm for a typical combinatorial optimization problem is given static input, and finds a single solution. Incremental optimization is one framework for handling problems that require solutions to be built up over time due to evolving constraints. An incremental algorithm is given a sequence of input, and finds a sequence of solutions that build incrementally while adapting to the changes in the input. Online algorithms also take in a sequence of input and produce incremental solutions; unlike their incremental counterparts, however, they are not given the input sequence in advance. Thus two factors can affect the competitive ratio of an online problem: that it does not know the input sequence in advance, and that it makes irreversible decisions. Both factors contribute to an algorithm's performance, but the competitive ratio fails to discern which is more significant. Incremental problems are only affected by the second factor, however, thus we can use incremental results to better understand online algorithms and improve their performance.
Given a graph G = (V, E), a (d, k)-coloring is a function from the vertices V to colors {1, 2, . ... more Given a graph G = (V, E), a (d, k)-coloring is a function from the vertices V to colors {1, 2, . . . , k} such that any two vertices within distance d of each other are assigned different colors. We determine the complexity of the (d, k)-coloring problem for all d and k, and enumerate some interesting properties of (d, k)-colorable graphs. Our main result is the discovery of a dichotomy between polynomial and NP-hard instances: for fixed d ≥ 2, the distance coloring problem is polynomial time for k ≤ 3d 2 and NP-hard for k > 3d 2 .
Traditional optimization algorithms are concerned with static input, static constraints, and atte... more Traditional optimization algorithms are concerned with static input, static constraints, and attempt to produce static output of optimal value. Recent literature has strayed from this conventional approach to deal with more realistic situations in which the input changes over time. Incremental optimization is a new framework for handling this type of dynamic behavior. We consider a general model for producing incremental versions of traditional covering problems along with several natural incremental metrics. Using this model, we demonstrate how to convert conventional algorithms into incremental algorithms with only a constant factor loss in approximation power. We introduce incremental versions of min cut, edge cover, and (k, r)-center and present some hardness results. Lastly, we discuss how the incremental model can help us more fully understand online problems and their corresponding algorithms.
Many combinatorial optimization problems aim to select a subset of elements of maximum value subj... more Many combinatorial optimization problems aim to select a subset of elements of maximum value subject to certain constraints. We consider an incremental version of such problems, in which some of the constraints rise over time. A solution is a sequence of feasible solutions, one for each time step, such that later solutions build on earlier solutions incrementally. We introduce a general model for such problems, and define incremental versions of maximum flow, bipartite matching, and knapsack. We find that imposing an incremental structure on a problem can drastically change its complexity. With this in mind, we give general yet simple techniques to adapt algorithms for optimization problems to their respective incremental versions, and discuss tightness of these adaptations with respect to the three aforementioned problems.
This paper defines an incremental version of the maximum flow problem. In this model, the capacit... more This paper defines an incremental version of the maximum flow problem. In this model, the capacities increase over time and the resulting solution is a sequence of flows that build on each other incrementally. Thus far, incremental problems considered in the literature have been built on NP-complete problems. To the best of our knowledge, our results are the first to find a polynomial time problem whose incremental version is NP-complete. We present approximation algorithms and hardness results for many versions of this problem, and comment on the relation to multicommodity flow.
Uploads
Papers by Alexa Sharp