On the Continuous CNN Problem

In the (discrete) CNN problem, online requests appear as points in $\mathbb{R}^2$. Each request must be served before the next one is revealed. We have a server that can serve a request simply by aligning either its $x$ or $y$ coordinate with the request. The goal of the online algorithm is to minimize the total $L_1$ distance traveled by the server to serve all the requests. The best known competitive ratio for the discrete version is 879 (due to Sitters and Stougie). We study the continuous version, in which, the request can move continuously in $\mathbb{R}^2$ and the server must continuously serve the request. A simple adversarial argument shows that the lower bound on the competitive ratio of any online algorithm for the continuous CNN problem is 3. Our main contribution is an online algorithm with competitive ratio $3+2 \sqrt{3} \approx 6.464$. Our analysis is tight. The continuous version generalizes the discrete orthogonal CNN problem, in which every request must be $x$ or $y$ aligned with the previous request. Therefore, Our result improves upon the previous best competitive ratio of 9 (due to Iwama and Yonezawa)

Optimal Power-Down Strategies

We consider the problem of selecting threshold times to transition a device to low-power sleep states during an idle period. The two-state case, in which there is a single active and a single sleep state, is a continuous version of the ski-rental problem. We consider a generalized version in which there is more than one sleep state, each with its own power-consumption rate and transition costs. We give an algorithm that, given a system, produces a deterministic strategy whose competitive ratio is arbitrarily close to optimal. We also give an algorithm to produce the optimal online strategy given a system and a probability distribution that generates the length of the idle period. We also give a simple algorithm that achieves a competitive ratio of $3 + 2\sqrt{2} \approx 5.828$ for any system

Enforcing efficient equilibria in network design games via subsidies

The efficient design of networks has been an important engineering task that involves challenging combinatorial optimization problems. Typically, a network designer has to select among several alternatives which links to establish so that the resulting network satisfies a given set of connectivity requirements and the cost of establishing the network links is as low as possible. The Minimum Spanning Tree problem, which is well-understood, is a nice example. In this paper, we consider the natural scenario in which the connectivity requirements are posed by selfish users who have agreed to share the cost of the network to be established according to a well-defined rule. The design proposed by the network designer should now be consistent not only with the connectivity requirements but also with the selfishness of the users. Essentially, the users are players in a so-called network design game and the network designer has to propose a design that is an equilibrium for this game. As it is usually the case when selfishness comes into play, such equilibria may be suboptimal. In this paper, we consider the following question: can the network designer enforce particular designs as equilibria or guarantee that efficient designs are consistent with users' selfishness by appropriately subsidizing some of the network links? In an attempt to understand this question, we formulate corresponding optimization problems and present positive and negative results.Comment: 30 pages, 7 figure

Robust Leader Election in a Fast-Changing World

We consider the problem of electing a leader among nodes in a highly dynamic network where the adversary has unbounded capacity to insert and remove nodes (including the leader) from the network and change connectivity at will. We present a randomized Las Vegas algorithm that (re)elects a leader in O(D\log n) rounds with high probability, where D is a bound on the dynamic diameter of the network and n is the maximum number of nodes in the network at any point in time. We assume a model of broadcast-based communication where a node can send only 1 message of O(\log n) bits per round and is not aware of the receivers in advance. Thus, our results also apply to mobile wireless ad-hoc networks, improving over the optimal (for deterministic algorithms) O(Dn) solution presented at FOMC 2011. We show that our algorithm is optimal by proving that any randomized Las Vegas algorithm takes at least omega(D\log n) rounds to elect a leader with high probability, which shows that our algorithm yields the best possible (up to constants) termination time.Comment: In Proceedings FOMC 2013, arXiv:1310.459

Evaluation of spatially dependent on-site detention basin policies

Includes bibliographical references.2015 Fall.Stormwater detention basins are typically used for stormwater control in many communities across the United States. They are commonly constructed downstream of every new development to control post-development runoff, and are called "on-site" detention basins. It has been shown by multiple authors in the literature that the design of on-site detention basins with no consideration of their location (non-spatially dependent policies, or Non-SD) in the watershed can actually increase peak flows above post-development peaks that would occur in the absence of on-site detention basins. This is caused by on-site detention basins delaying the peak release of a particular subwatershed and combining with other peak flows in the watershed (McCuen 1974; McCuen 1979; Emerson et al. 2005). Strategies to combat this problem have been reported, but metrics used to judge their success are limited to the main channel of the watershed or the watershed outlet only, leaving its impact in the remaining other watershed locations unknown. In addition, some strategies have recommended increasing the storage of on-site detention basins, but this approach would increase construction and maintenance costs and reduce the amount of land available to developers. Validation of increased peak flows throughout the watershed when Non-SD policies are used to design on-site detention basins compared to no on-site detention in the watershed was investigated first. The Non-SD policies used in this study controlled the post-development 10 and 100-year peak flows to flows at or below their respective pre-development peak flows (Non-SD 1), and controlled the post-development 100-year peak flow to flows at or below the 2-year pre-development peak flow (Non-SD 2). Next, spatially dependent policies (SD policies) were created by altering the peak flow release from on-site detention basins that would have occurred under a Non-SD policy based on its location in the watershed. These peak flows were altered using a linear model and a piece-wise linear model. Results from SD policies were compared to those from Non-SD policies. Metrics used to evaluate the effectiveness of the on-site detention basin policies (both SD and Non-SD) were peak flows throughout the watershed and total watershed storage. All policies were tested on a watershed in Fort Collins, Colorado using the Urban Morpho-climatic Instantaneous Unit Hydrograph model. Results indicate that Non-SD polices effectively reduce peak flows throughout the watershed, and do not increase peak flows compared to a policy that uses no on-site detention. When compared against Non-SD 1, SD policies derived from the linear equation were successful at reducing peak flows at some 2nd and 3rd order channel and pipe intersections in the upper half of the watershed, while increasing peak flows at 2nd order channel and pipe intersections in the lower half of the watershed. The remaining intersections were not effected by this SD policy, and the total watershed storage was shown to increase. SD policies derived from the piece-wise linear model increased peak flows at 2nd order channel and pipe intersections in the lower half of the watershed. The remaining intersections were not affected by this SD policy, and watershed storage was shown to slightly decrease. When compared to Non-SD 2, SD policies had little to no effect on peak flows at any location in the watershed or on the watershed storage

Offline and online variants of the Traveling Salesman Problem

In this thesis, we study several well-motivated variants of the Traveling Salesman Problem (TSP). First, we consider makespan minimization for vehicle scheduling problems on trees with release and handling times. 2-approximation algorithms were known for several variants of the single vehicle problem on a path. A 3/2-approximation algorithm was known for the single vehicle problem on a path where there is a fixed starting point and the vehicle must return to the starting point upon completion. Karuno, Nagamochi and Ibaraki give a 2-approximation algorithm for the single vehicle problem on trees. We develop a Polynomial Time Approximation Scheme (PTAS) for the single vehicle scheduling problem on trees which have a constant number of leaves. This PTAS can be easily adapted to accommodate various starting/ending constraints. We then extended this to a PTAS for the multiple vehicle problem where vehicles operate in disjoint subtrees. We also present competitive online algorithms for some single vehicle scheduling problems. Secondly, we study a class of problems called the Online Packet TSP Class (OP-TSP-CLASS). It is based on the online TSP with a packet of requests known and available for scheduling at any given time. We provide a 5/3 lower bound on any online algorithm for problems in OP-TSP-CLASS. We extend this result to the related k-reordering problem for which a 3/2 lower bound was known. We develop a κ+1-competitive algorithm for problems in OP-TSP-CLASS, where a κ-approximation algorithm is known for the offline version of that problem. We use this result to develop an offline m(κ+1)-approximation algorithm for the Precedence-Constrained TSP (PCTSP) by segmenting the n requests into m packets. Its running time is mf(n/m) given a κ-approximation algorithm for the offline version whose running time is f(n)