A note on: No need to choose: How to get both a PTAS and Sublinear Query Complexity

Abstract

We revisit various PTAS's (Polynomial Time Approximation Schemes) for minimization versions of dense problems, and show that they can be performed with sublinear query complexity. This means that not only do we obtain a (1 + ε)-approximation to the NP-Hard problems in polynomial time, but also avoid reading the entire input. This setting is particularly advantageous when the price of reading parts of the input is high, as is the case, for examples, where humans provide the input. Trading off query complexity with approximation is the raison d'etre of the field of learning theory, and of the ERM (Empirical Risk Minimization) setting in particular. A typical ERM result, however, does not deal with computational complexity. We discuss two particular problems for which (a) it has already been shown that sublinear querying is sufficient for obtaining a (1 + ε)-approximation using unlimited computational power (an ERM result), and (b) with full access to input, we could get a (1 + ε)-approximation in polynomial time (a PTAS). Here we show that neither benefit need be sacrificed. We get a PTAS with efficient query complexity. The first problem is known as Minimal Feedback Arc-Set in Tournaments (MFAST). A PTAS has been discovered by Schudy and Mathieu, and an ERM result by Ailon. The second is k-Correlation Clustering (k-CC). A PTAS has been discovered by Giotis and Guruswami, and an ERM result by Ailon and Begleiter. Two techniques are developed. The first solves the problem for the lowcost case of k-CC (the analogous case is already known for MFAST). This requires a careful sampling scheme together with proof of a structural property relating costs of vertices against the optimal sample clustering with their costs against the full optimal clustering. The second addresses the high-cost case, by showing that a classic method by Arora et al. (2002) for obtaining additive approximations can be made query efficient. The underlying technique is "double sampling": One sample is amenable to exhaustive solution enumeration, but well approximates only polynomially many solutions (including the optimal), and another sample cannot be used exhaustively search solutions, but well approximates the cost of the enitre solution space, and is used for verification.

Sublinear Algorithms for MAXCUT and Correlation Clustering

“…In this paper, we focus on the maximum-agreement variant of correlation clustering and in particular we focus on (1 + ǫ)-approximations. Here, Ailon and Karnin [5] presented an approximation scheme with sublinear query complexity (which also yields a semi-streaming algorithm) for dense instances of correlation clustering. Giotis and Guruswami [19] described a sampling based algorithm combined with a greedy strategy which guarantees a solution within (ǫn 2 ) additive error.…”