An official website of the United States government
Vizing’s celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2∆−1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to lowdegree graphs, with ∆ = O(log n), and they conjectured the existence of online algorithms using ∆(1 + o(1)) colors for ∆ = ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))∆-edge-coloring algorithm for ∆ = ω(log n) known a priori. Surprisingly, if ∆ is not known ahead of time, we show that no (e/(e−1)−Ω(1))∆-edge-coloring algorithm exists.We then provide an optimal, (e/(e−1) +o(1))∆-edge-coloring algorithm for unknown ∆ = ω(log n). Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.
more »
« less
The Average-Value Allocation Problem
We initiate the study of centralized algorithms for welfare-maximizing allocation of goods to buyers subject to average-value constraints. We show that this problem is NP-hard to approximate beyond a factor of e/(e-1), and provide a 4e/(e-1)-approximate offline algorithm. For the online setting, we show that no non-trivial approximations are achievable under adversarial arrivals. Under i.i.d. arrivals, we present a polytime online algorithm that provides a constant approximation of the optimal (computationally-unbounded) online algorithm. In contrast, we show that no constant approximation of the ex-post optimum is achievable by an online algorithm.
more »
« less
- PAR ID:
- 10555430
- Editor(s):
- Kumar, Amit; Ron-Zewi, Noga
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 317
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-348-5
- Page Range / eLocation ID:
- 317-317
- Subject(s) / Keyword(s):
- Resource allocation return-on-spend constraint approximation algorithm online algorithm Theory of computation → Approximation algorithms analysis Theory of computation → Online algorithms
- Format(s):
- Medium: X Size: 23 pages; 927400 bytes Other: application/pdf
- Size(s):
- 23 pages 927400 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We study a cache network under arbitrary adversarial request arrivals. We propose a distributed online policy based on the online tabular greedy algorithm. Our distributed policy achieves sublinear (1-1/e)-regret, also in the case when update costs cannot be neglected. Numerical evaluation over several topologies supports our theoretical results and demonstrates that our algorithm outperforms state-of-art online cache algorithms.more » « less
-
We provide an approximation algorithm for network revenue management problems. In our approximation algorithm, we construct an approximate policy using value function approximations that are expressed as linear combinations of basis functions. We use a backward recursion to compute the coefficients of the basis functions in the linear combinations. If each product uses at most L resources, then the total expected revenue obtained by our approximate policy is at least [Formula: see text] of the optimal total expected revenue. In many network revenue management settings, although the number of resources and products can become large, the number of resources used by a product remains bounded. In this case, our approximate policy provides a constant-factor performance guarantee. Our approximate policy can handle nonstationarities in the customer arrival process. To our knowledge, our approximate policy is the first approximation algorithm for network revenue management problems under nonstationary arrivals. Our approach can incorporate the customer choice behavior among the products, and allows the products to use multiple units of a resource, while still maintaining the performance guarantee. In our computational experiments, we demonstrate that our approximate policy performs quite well, providing total expected revenues that are substantially better than its theoretical performance guarantee.more » « less
-
Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay delta is incurred during which no data can be transmitted. For the offline version of the problem we present a (1-(1/e)-epsilon) approximation ratio (for any constant epsilon >0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e-1)/(2e-1)-epsilon)-competitive ratio (for any constant epsilon >0 ) that exceeds time by an additive factor of O(delta/epsilon). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one.more » « less
-
The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging, and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal deterministic competitive ratio of 1/2 for either of these models was open. In this paper, we resolve this question for both these natural arrival models, and show the following. For edge arrivals, randomization does not help | no randomized algorithm is better than 1/2 competitive. For general vertex arrivals, randomization helps | there exists a randomized (1/2+ Ω(1))-competitive online matching algorithm.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.13
