Low congestion online routing and an improved mistake bound for online prediction of graph labeling

arXiv:0809.2075v2 [cs.DS] 12 Sep 2008 Low congestion online routing and an improved mistake bound for online prediction of graph labeling Jittat Fakcharoenphol∗ Boonserm Kijsirikul† Abstract In this paper, we show a connection between a certain online low-congestion routing problem and an online prediction of graph labeling. More specifically, we prove that if there exists a routing scheme that guarantees a congestion of α on any edge, there exists an online prediction algorithm with mistake bound α times the cut size, which is the size of the cut induced by the label partitioning of graph vertices. With previous known bound of O(log n) for α for the routing problem on trees with n vertices, we obtain an improved prediction algorithm for graphs with high effective resistance. In contrast to previous approaches that move the graph problem into problems in vector space using graph Laplacian and rely on the analysis of the perceptron algorithm, our proof are purely combinatorial. Further more, our approach directly generalizes to the case where labels are not binary. 1 Introduction We are interested in an online prediction problem on graphs. Given a connected graph G = (V, E) and a labeling ℓ: V →{−1, +1}, unknown to the prediction algorithm, in each round i, for i = 1, 2, . . ., an adversary asks for a label of a vertex vi ∈V , the prediction algorithm provides the answer yi, and then receives the correct label ˆyi = ℓ(vi). The goal is to minimize the number of rounds that the algorithm makes a mistake, i.e., rounds i such that yi ̸= ˆyi. To make our presentation clean, in this work we do not count the mistake made on the first question v1.1 This problem has been studied with standard online learning tools such as the perceptron algorithm. Herbster, Pontil, and Wainer [6], and Herbster and Pontil [5] use pseudoinverse of graph Laplacian as a kernel and provide a mistake bound that depends on the size of the cut induced by the partition based on the real labeling of vertices and the largest effective resistance between any pair of vertices in the graph. Recently, Herbster [4] exploits the cluster structure of the labeling on the graph, and provides an improved mistake bounds. Pelckmans and Suykens [7] present a combinatorial algorithm for the problem that predicts a label of a given vertex based on known labels of its neighbors. They also prove a bound on the number of mistakes when the labels of adjacent vertices are known. However, their bound is very loose since it does not count every mistakes and their proof is still based on graph Laplacian. We shall compare the bound that we obtain with previous bounds of Herbster et. al. [6, 5, 4] and of Pelckmans and Suykens [7] in Section 3.1. ∗Department of Computer Engineering, Kasetsart University, Bangkok, Thailand 10900. E-mail: jittat@gmail.com. Supported by the Thailand Research Fund Grant MRG5080318. †Department of Computer Engineering, Chulalongkorn University, Bangkok, Thailand 10330. E-mail: Boonserm.K@chula.ac.th. 1To properly account this, one can simply add 1 to our mistake bound. This work follows the initiation of Pelckmans and Suykens. We show connection between the prediction problem and the following online routing problem, first introduced by Awerbuch and Azar [1] in their study of online multicast routing. Given a connected graph G = (V, E), the algorithm receives a sequence of requests r1, r2, . . ., where ri ∈V , and, for each ri, where i > 0, has to route one unit of flow from ri to some previous know rj where j < i. The algorithm works in an online fashion, i.e., it has to return a route for ri before receiving other requests ri′, where i′ > i. Given a set of routes, we define the congestion Cong(e) incurred on edge e ∈E, defined as the number of routes that use e. The performance of the algorithm is measured by the maximum congestion incurred on any edge. We prove, in Section 2, that if there exists an algorithm A with a guarantee that the congestion incurred on any edge will be no greater than α, there exists an online prediction algorithm with the mistake bound of α · |cut(ℓ)|, where cut(ℓ) be the set of edges joining pairs of vertices with different labels, i.e., cut(ℓ) = {(u, v) ∈E : ℓ(u) ̸= ℓ(v)}. In Section 3, we apply the known congestion bound to show the mistake bound for the graph prediction problem, and compare the bound obtained with the bounds from previous results. We note that our approach directly generalizes to the case when labels are not binary (i.e., when the labeling function ℓmaps V to an arbitrary set L of labels) with the same mistake bound. 2 Reduction to low-congestion routing We first present an online prediction algorithm from an online routing algorithm A. The pre- diction algorithm PA is very simple, given a vertex vi, it uses A to route one unit of flow from vi to any vertices vj with known labels, it then returns the known label ℓ(vj) as the prediction. We prove the following theorem. Theorem 1 If A guarantees that no

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