Maximizing Rides Served for Dial-a-Ride on the Uniform Metric

In this paper, we present tight bounds for the k-server problem with delays in the uniform metric space. The problem is defined on n+k nodes in the uniform metric space which can issue requests over time. These requests can be served directly or with some delay using k servers, by moving a server to the corresponding node with an open request. The task is to find an online algorithm that can serve the requests while minimizing the total moving and delay costs. We first provide a lower bound by showing that the competitive ratio of any deterministic online algorithm cannot be better than (2k+1) in the clairvoyant setting. We will then show that conservative algorithms (without delay) can be equipped with an accumulative delay function such that all such algorithms become (2k+1)-competitive in the non-clairvoyant setting. Together, the two bounds establish a tight result for both, the clairvoyant and the non-clairvoyant settings.

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of 1- 1/e ¿ 0.632, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the 1 - 1/e bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this 1 - ¿ barrier. Furthermore, we show that no online algorithm can produce a 1 - ¿ approximation for an arbitrarily small e for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.

We propose a general method for converting online algorithms to local computation algorithms, by selecting a random permutation of the input, and simulating running the online algorithm. We bound the number of steps of the algorithm using a query tree, which models the dependencies between queries. We improve previous analyses of query trees on graphs of bounded degree, and extend this improved analysis to the cases where the degrees are distributed binomially, and to a special case of bipartite graphs. Using this method, we give a local computation algorithm for maximal matching in graphs of bounded degree, which runs in time and space O(log3 n). We also show how to convert a large family of load balancing algorithms (related to balls and bins problems) to local computation algorithms. This gives several local load balancing algorithms which achieve the same approximation ratios as the online algorithms, but run in O(logn) time and space. Finally, we modify existing local computation algorithms for hypergraph 2-coloring and k-CNF and use our improved analysis to obtain better time and space bounds, of O(log4 n), removing the dependency on the maximal degree of the graph from the exponent.

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

We investigate a generalization of classical shop scheduling where n jobs are located at the vertices of a general undirected graph and m machines must travel between the vertices to process the jobs. The aim is to minimize the makespan. For the open shop problem, we develop an O(logmloglogm)-approximation algorithm that significantly improves upon the best known $O(\sqrt{m})$ -approximation algorithm. For the flow shop problem, we present an O(m 2/3)-approximation algorithm that improves upon the best known $\max\{\frac{m+1}{2},\rho\}$ -approximation algorithm, where ρ is the approximation factor for metric TSP.

During difficult foraging tasks, humans rarely switch between target categories, but switch frequently during easier foraging. Does this reflect fundamental limits on visual working memory (VWM) capacity or simply strategic choice due to effort? Our participants performed time-limited or unlimited foraging tasks where they tapped stimuli from 2 target categories while avoiding items from 2 distractor categories. These time limits should have no effect if capacity imposes limits on VWM representations but more flexible VWM could allow observers to use VWM according to task demands in each case. We found that with time limits, participants switched more frequently and switch-costs became much smaller than during unlimited foraging. Observers can therefore switch between complex (conjunction) target categories when needed. We propose that while maintaining many complex templates in working memory is effortful and observers avoid this, they can do so if this fits task demands, showing the flexibility of working memory representations used for visual exploration. This is in contrast with recent proposals, and we discuss the implications of these findings for theoretical accounts of working memory. (PsycINFO Database Record

Input Queued (IQ) switch architecture has been of interest due to its low memory bandwidth requirement. A scheduling algorithm is required to schedule the transfer of packets through cross-bar switch fabric at every time slot. The performance, that is throughput and delay, of a switch depends on the scheduling algorithm. The maximum weight matching (MWM) algorithm is known to deliver 100% throughput under any admissible traffic. Leonardi et. al. (2001) obtained a nontrivial bound on the delay for the MWM algorithm under admissible Bernoulli i.i.d. traffic. There has been a lot of interesting work done over time to analyze throughput of scheduling algorithms. But apart from the work of Leonardi et al. there has not been any work done to obtain bounds on delay of scheduling algorithms. The MWM algorithm is perceived to be a very good scheduling algorithm in general and simulations have suggested that it performs better than most of the known algorithms in terms of delay. But it is very complex to implement. Hence many simple to implement approximations to MWM have been proposed. In this paper, we study a class of algorithms to MWM, which always obtain a schedule whose weight W differs from the weight of MWM schedule W* by at most f(W*), where f(.) is a sub-linear function. We call this difference in weight as approximation distance of algorithm from MWM. We denote this class of algorithms by 1-APRX. We prove that any 1-APRX algorithm is stable, that is, it delivers upto 100% of throughput under any admissible Bernoulli i.i.d. input traffic. Under any admissible Bernoulli i.i.d. traffic, we obtain bounds on the average queue length(equivalently delay) of the 1-APRX algorithms using a Lyapunov function technique, which was motivated in Leonardi et al. The delay bounds obtained for the 1-APRX algorithm are linearly related with the approximation distance, which matches the intuition that the better the weight of the schedule, the better the algorithm will perform. Interestingly, simulations show a linear relationship between the average queue length (equivalently delay) and the approximation distance. Thus, the approximation distance of a scheduling algorithm can serve as a metric to differentiate between the performance of different stable algorithms, even though throughput may be same for these algorithms. We also obtain a novel heuristic tighter bound on the average queue length (equivalently delay) under uniform Bernoulli i.i.d. traffic for MWM using a very simple argument.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

We consider a variant of the offline Dial-a-Ride problem with a single server where each request has a source, destination, and a prize earned for serving it. The goal for the server is to serve requests within a given time limit so as to maximize the total prize money. We consider the variant where prize amounts are uniform which is equivalent to maximizing the number of requests served. This setting is applicable when all rides may have equal importance such as paratransit services. We first prove that no polynomial-time algorithm can be guaranteed to serve the optimal number of requests, even when the time limit for the algorithm is augmented by any constant factor \(c \ge 1\). We also show that if \(\lambda = t_{max}/t_{min}\), where \(t_{max}\) and \(t_{min}\) denote the largest and smallest edge weights in the graph, the approximation ratio for a reasonable class of algorithms for this problem is unbounded, unless \(\lambda \) is bounded. We then show that the segmented best path (\(\textsc {sbp}\)) algorithm from [8] is a 4-approximation. We then present our main result, an algorithm, k-Sequence, that repeatedly serves the fastest set of k remaining requests, and provide upper and lower bounds on its performance. We show k-Sequence has approximation ratio at most \(2+\lceil \lambda \rceil /k\) and at least \(1 + \lambda /k\) and that \(1 + \lambda /k\) is tight when \(1 + \lambda /k \ge k\). Thus, for the case of \(k=1\), i.e., when the algorithm repeatedly serves the quickest request, it has approximation ratio \(1+\lambda \), which is tight for all \(\lambda \). We also show that even as k grows beyond the size of \(\lambda \), the ratio never improves below 9/7.

We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those jobs. A strongly local policy only uses the processing time of jobs on the the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortest-first algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem, and answers an open question raised by Ibarra and Kim [16], and Davis and Jaffe [10]. We then design a local ordering policy with the approximation ratio of Θ(log m) in equilibria, and prove that this policy is optimal among all local ordering policies. This policy orders the jobs in the non-decreasing order of their inefficiency, i.e, the ratio between the processing time on that machine over the minimum processing time. Finally, we show that best responses of players for the inefficiency-based policy may not converge to a pure Nash equilibrium, and present a Θ(log2 m) policy for which we can prove fast convergence of best responses to pure Nash equilibria.

This paper investigates the existence of fixed points and best proximity points of p-cyclic self-maps on a set of subsets of a certain uniform space under integral-type contractive conditions. The parallel properties of the associated restricted composed maps from any of the subsets to itself are also investigated. The subsets of the uniform space are not assumed to intersect.

Graph Theory Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.

In this paper we introduce two general techniques for the design and analysis of approximation algorithms for 𝒩𝒫-hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times. For a variety of scheduling models, these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum. In the first approach, we use an optimal solution to a linear programming relaxation in order to guide a simple list-scheduling rule. Consequently, we also obtain results about the strength of the relaxation. Our second approach yields on-line algorithms for these problems: in this setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. Our on-line technique yields constant performance guarantees for a variety of scheduling environments, and in some cases essentially matches the performance of our off-line LP-based algorithms.

Mean analysis of an online algorithm for the vertex cover problem

We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. Like an online algorithm, a priority algorithm receives its input sequentially and must output irrevocable decisions for individual input items before having seen the entire input. However, in contrast to the online setting, a priority algorithm may choose an order on the set of all possible input items and the actual input is then presented according to this order. A priority algorithm is thus a natural model for the intuitively well-understood concept of a greedy algorithm.Mainly motivated by their application for proving lower bounds, we also consider priority algorithms with advice, thus measuring the necessary amount of information about the yet unknown parts of the input.Besides considering the classical variant of the DPA problem on paths and the related problem of Length-Weighted DPA, we mainly focus on DPA on trees. We show asymptotically matching upper and lower bounds on the advice necessary for optimality in LWDPA and generalize the known optimality result for DPA on paths to trees with maximal degree at most 3. On trees with higher maximal degree, we prove matching upper and lower bounds on the approximation ratio in the advice-free priority setting as well as upper and lower bounds on the advice necessary to achieve optimality.KeywordsDisjoint path allocationPriority algorithmsAdvice complexityGreedy algorithms