Achievable Competitive Ratio Research Articles

Competitive Online Path-Aware Path Selection

This paper studies an online path selection problem and proposes online mechanisms for a network operator to sequentially update link prices. The aim is to incentivize online-arriving agents to join the network and select paths in a manner that maximizes the social welfare, which comprises both system profit and the quality of service experienced by agents. Competitive analysis is adopted to analyze the performance of the proposed online mechanism, whose best achievable competitive ratio is 4. Sufficient and necessary conditions on a competitive mechanism are established. Moreover, the performance limit of the celebrated multiplethe- index pricing scheme is also analyzed.

A stronger impossibility for fully online matching

We revisit the fully online matching model (Huang et al., 2020), an extension of the classic online matching model due to Karp, Vazirani, and Vazirani (1990), which has recently received a lot of attention (Huang et al., 2019, 2020), partly due to applications in ride-sharing platforms. It has been shown that the fully online version is harder than the classic version for which the achievable competitive ratio is at most 0.6317, rather than precisely 1−1e≈0.6321. We introduce two new ideas to the construction. By optimizing the parameters of the modified construction numerically, we obtain an improved impossibility result of 0.6297. Like the previous bound, the new bound even holds for fractional (rather than randomized) algorithms on bipartite graphs.

On-line Search in Two-Dimensional Environment

We consider the following on-line pursuit-evasion problem. A team of mobile agents called searchers starts at an arbitrary node of an unknown network. Their goal is to execute a search strategy that guarantees capturing a fast and invisible intruder regardless of its movements using as few searchers as possible. We require that the strategy is connected and monotone, that is, at each point of the execution the part of the graph that is guaranteed to be free of the fugitive is connected and whenever some node gains a property that it cannot be occupied by the fugitive, the strategy must operate in such a way to keep this property till its end. As a way of modeling two-dimensional shapes, we restrict our attention to networks that are embedded into partial grids: nodes are placed on the plane at integer coordinates and only nodes at distance one can be adjacent. Agents do not have any knowledge about the graph a priori, but they recognize the direction of the incident edge (up, down, left or right). We give an on-line algorithm for the searchers that allows them to compute a connected and monotone strategy that guarantees searching any unknown partial grid with the use of O(sqrt {n}) searchers, where n is the number of nodes in the grid. As for a lower bound, there exist partial grids that require {varOmega }(sqrt {n}) searchers. Moreover, we prove that for each on-line searching algorithm there is a partial grid that forces the algorithm to use {varOmega }(sqrt {n}) searchers but O(log n) searchers are sufficient in the off-line scenario. This gives a lower bound on {varOmega }(sqrt {n}/log n) in terms of achievable competitive ratio of any on-line algorithm.

Online Minimum Spanning Tree with Advice

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.

Speed Scaling of Processes with Arbitrary Speedup Curves on a Multiprocessor

We consider the setting of a multiprocessor where the speeds of the m processors can be individually scaled. Jobs arrive over time and have varying degrees of parallelizability. A nonclairvoyant scheduler must assign the processes to processors, and scale the speeds of the processors. We consider the objective of energy plus flow time. We assume that a processor running at speed s uses power s α for some constant α>1. For processes that may have side effects or that are not checkpointable, we show an $\Omega(m^{(\alpha -1)/\alpha^{2}})$ bound on the competitive ratio of any randomized algorithm. For checkpointable processes without side effects, we give an O(log m)-competitive algorithm. Thus for processes that may have side effects or that are not checkpointable, the achievable competitive ratio grows quickly with the number of processors, but for checkpointable processes without side effects, the achievable competitive ratio grows slowly with the number of processors. We then show a lower bound of Ω(log 1/α m) on the competitive ratio of any randomized algorithm for checkpointable processes without side effects.

Nonclairvoyant Speed Scaling for Flow and Energy

We give three results related to online nonclairvoyant speed scaling to minimize total flow time plus energy. We give a nonclairvoyant algorithm LAPS, and show that for every power function of the form P(s)=s α, LAPS is O(1)-competitive; more precisely, the competitive ratio is 8 for α=2, 13 for α=3, and \(\frac{2\alpha^{2}}{\ln\alpha}\) for α>3. We then show that there is no constant c, and no deterministic nonclairvoyant algorithm A, such that A is c-competitive for every power function of the form P(s)=s α. So necessarily the achievable competitive ratio increases as the steepness of the power function increases. Finally we show that there is a fixed, very steep, power function for which no nonclairvoyant algorithm can be O(1)-competitive.

Improved online algorithms for the sorting buffer problem on line metrics

An instance of the sorting buffer problem consists of a metric space and a server, equipped with a finite-capacity buffer capable of holding a limited number of requests. An additional ingredient of the input is an online sequence of requests, each of which is characterized by a destination in the given metric space; whenever a request arrives, it must be stored in the sorting buffer. At any point in time, a currently pending request can be served by drawing it out of the buffer and moving the server to its corresponding destination. The objective is to serve all input requests in a way that minimizes the total distance traveled by the server. In this article, we focus our attention on instances of the problem in which the underlying metric is either an evenly-spaced line metric or a continuous line metric . Our main findings can be briefly summarized as follows. (1) We present a deterministic O (log n )-competitive algorithm for n -point evenly-spaced line metrics. This result improves on a randomized O (log 2 n )-competitive algorithm due to Khandekar and Pandit [2006b]. It also refutes their conjecture, stating that a deterministic strategy is unlikely to obtain a nontrivial competitive ratio. (2) We devise a deterministic O (log N log log N )-competitive algorithm for continuous line metrics, where N denotes the length of the input sequence. In this context, we introduce a novel discretization technique of independent interest. (3) We establish the first nontrivial lower bound for the evenly-spaced case, by proving that the competitive ratio of any deterministic algorithm is at least 2 + √3/√3 ≈ 2.154. This result settles, to some extent, an open question due to Khandekar and Pandit [2006b], who posed the task of attaining lower bounds on the achievable competitive ratio as a foundational objective for future research.

On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart from the packing dynamically. The main result in this paper is a lower bound of 2.5 on the achievable competitive ratio, improving the best known 2.428 lower bound, and revealing that packing items of restricted form like unit fractions (i.e., of size 1/k for some integer k), for which a 2.4985-competitive algorithm is known, is indeed easier. We also investigate the resource augmentation version of the problem where the on-line algorithm can use bins of size b (>1) times that of the optimal off-line algorithm. An interesting result is that we prove b=2 is both necessary and sufficient for the on-line algorithm to match the performance of the optimal off-line algorithm, i.e., achieve 1-competitiveness. Further analysis gives a trade-off between the bin size multiplier 1<b≤2 and the achievable competitive ratio.