Online algorithm | EPFL Graph Search

In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. In operations research, the area in which online algorithms are developed is called online optimization. As an example, consider the sorting algorithms selection sort and insertion sort: selection sort repeatedly selects the minimum element from the unsorted remainder and places it at the front, which requires access to the entire input; it is thus an offline algorithm. On the other hand, insertion sort considers one input element per iteration and produces a partial solution without considering future elements. Thus insertion sort is an online algorithm. Note that the final result of

How to Match in Modern Computational Settings

Buddhima Ruwanmini Gamlath Gamlath Ralalage

This thesis focuses on the maximum matching problem in modern computational settings where the algorithms have to make decisions with partial information.First, we consider two stochastic models called query-commit and price-of-information where the algorithm only knows the distribution from which the edges are sampled.In the query-commit model, the algorithm must query edges to know if they exist and is committed to adding all queried edges that exist to its output.In the price-of-information model, the algorithm incurs costs for querying edges, and the total query cost is subtracted from the output matching's weight.For maximum weighted matching in these models, previously known best algorithms were greedy algorithms that achieve 1/2 approximations. We improve the approximation ratio to 1 - 1/e in both models. Next, we consider situations where the input graphs do not fit into the space available for an algorithm instance. We consider two such models: the semi-streaming model where the algorithm receives the input as a stream of edges and the algorithm has only sub-linear (in the number of edges) space, and the massively parallel computation (MPC) model where the input is distributed among several machines, each of which has sub-linear space, and algorithm instances running on different machines must communicate in synchronous rounds.We start with a particular case of the semi-streaming model where the edges arrive in uniformly random order, and the algorithm goes over the stream only once. For this setting, we give the first algorithm that finds a (1/2 + c)-approximate maximum weighted matching in expectation; such algorithms were previously known only for the unweighted graphs.We then show how to efficiently find (1 - epsilon)-approximate weighted matchings for any epsilon > 0 in multi-pass semi-streaming and MPC models by extending our algorithmic ideas used in the single-pass semi-streaming model with random order edge arrivals.Finally, we study online algorithms for matching, where the input graph is gradually revealed over time. In the online edge-arrival setting, the graph is revealed one edge at a time, and an algorithm is forced to make irrevocable decisions on whether to add each edge to the output matching upon their arrival. We show that no online algorithm can achieve a competitive ratio of 1/2 + c for any constant c > 0 in this setting.In the online vertex-arrival setting, the graph is revealed one vertex at a time, together with its incident edges to already revealed vertices, and the algorithm must irrevocably decide to ignore the revealed vertex or match it to one of the available neighbors.In this setting, we show how to round a previously known fractional online matching algorithm to get an integral online matching algorithm with a competitive ratio of 1/2 + c for some constant c > 0.

EPFL2021

Tight Bounds for Online Edge Coloring

David Wajc

Vizing's celebrated theorem asserts that any graph of maximum degree Delta admits an edge coloring using at most Delta + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2 Delta - 1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with Delta = O(log n), and they conjectured the existence of online algorithms using Delta (1+o(1)) colors for Delta = omega(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))Delta-edge-coloring algorithm for Delta = omega(log n) known a priori. Surprisingly, if Delta is not known ahead of time, we show that no (e/e-1 - Omega(1)) Delta-edge-coloring algorithm exists. We then provide an optimal, (e/e-1 + o(1))Delta-edge-coloring algorithm for unknown Delta = omega(log n). To obtain our results, we study a nonstandard 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.

IEEE COMPUTER SOC2019

An online throughput-competitive algorithm for multicast routing and admission control

Ashish Goel, Monika Henzinger

We present the first polylog-competitive online algorithm for the general multicast admission control and routing problem in the throughput model. The ratio of the number of requests accepted by the optimum offline algorithm to the expected number of requests accepted by our algorithm is O((logn+loglogM) (logn+logM)logn), where M is the number of multicast groups and n is the number of nodes in the graph. We show that this is close to optimum by presenting an 0(lognlogM) lower bound on this ratio for any randomized online algorithm against an oblivious adversary, when M is much larger than the link capacities. Our lower bound applies even in the restricted case where the link capacities are much larger than bandwidth requested by a single multicast. We also present a simple proof showing that it is impossible to be competitive against an adaptive online adversary. As in the previous online routing algorithms, our algorithm uses edge-costs when deciding on which is the best path to use. In contrast to the previous competitive algorithms in the throughput model, our cost is not a direct function of the edge load. The new cost definition allows us to decouple the effects of routing and admission decisions of different multicast groups.

2005

How to Match in Modern Computational Settings

Buddhima Ruwanmini Gamlath Gamlath Ralalage

EPFL2021