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We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently [3,5,6,2]. The basic setup here is that we are given as input a complete graph on n nodes (which correspond to nodes to be clustered) whose edges are labeled + (for similar pairs of items) and − (for dissimilar pairs of items). Thus we have only as input qualitative information on similarity and no quantitative distance measure between items. The quality of a clustering is measured in terms of its number of agreements, which is simply the number of edges it correctly classifies, that is the sum of number of − edges whose endpoints it places in different clusters plus the number of + edges both of whose endpoints it places within the same cluster.In this paper, we study the problem of finding clusterings that maximize the number of agreements, and the complementary minimization version where we seek clusterings that minimize the number of disagreements. We focus on the situation when the number of clusters is stipulated to be a small constant k. Our main result is that for every k, there is a polynomial time approximation scheme for both maximizing agreements and minimizing disagreements. (The problems are NP-hard for every k ≥ 2.) The main technical work is for the minimization version, as the PTAS for maximizing agreements follows along the lines of the property tester for Max k-CUT from [11].In contrast, when the number of clusters is not specified, the problem of minimizing disagreements was shown to be APX-hard [5], even though the maximization version admits a PTAS.

How should players bid in keyword auctions such as those used by Google, Yahoo! and MSN? We consider greedy bidding strategies for a repeated auction on a single keyword, where in each round, each player chooses some optimal bid for the next round, assuming that the other players merely repeat their previous bid. We study the revenue, convergence and robustness properties of such strategies. Most interesting among these is a strategy we call the balanced bidding strategy (bb): it is known that bb has a unique fixed point with payments identical to those of the VCG mechanism. We show that if all players use the bb strategy and update each round, bb converges when the number of slots is at most 2, but does not always converge for 3 or more slots. On the other hand, we present a simple variant which is guaranteed to converge to the same fixed point for any number of slots. In a model in which only one randomly chosen player updates each round according to the bb strategy, we prove that convergence occurs with probability 1. We complement our theoretical results with empirical studies.

We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree ∆ has an acyclic proper edge coloring with at most ⌈3.74(∆ − 1)⌉ + 1 colors, whereas, previously, the best bound was 4(∆ − 1). The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

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