Social networks involve both positive and negative relationships, which can be captured in signed graphs. The edge sign prediction problem aims to predict whether an interaction between a pair of nodes will be positive or negative. We provide theoretical results for this problem that motivate natural improvements to recent heuristics.The edge sign prediction problem is related to correlation clustering; a positive relationship means being in the same cluster. We consider the following model for two clusters: we are allowed to query any pair of nodes whether they belong to the same cluster or not, but the answer to the query is corrupted with some probability 0 < q < 1 2 . Let δ = 1 − 2q be the bias. We provide an algorithm that recovers all signs correctly with high probability in the presence of noise with O( n log n δ 2 + log 2 n δ 6 ) queries. This is the best known result for this problem for all but tiny δ, improving on the recent work of Mazumdar and Saha [27]. We also provide an algorithm that performs O( n log n δ 4 ) queries, and uses breadth first search as its main algorithmic primitive. While both the running time and the number of queries for this algorithm are sub-optimal, our result relies on novel theoretical techniques, and naturally suggests the use of edge-disjoint paths as a feature for predicting signs in online social networks. Correspondingly, we experiment with using edge disjoint s − t paths of short length as a feature for predicting the sign of edge (s, t) in real-world signed networks. Empirical findings suggest that the use of such paths improves the classification accuracy, especially for pairs of nodes with no common neighbors.

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Subset Sum Ratio is the following optimization problem: Given a set of n positive numbers I, find disjoint subsets X, Y Ď I minimizing the ratio maxtΣpXq{ΣpY q, ΣpY q{ΣpXqu, where ΣpZq denotes the sum of all elements of Z. Subset Sum Ratio is an optimization variant of the Equal Subset Sum problem. It was introduced by Woeginger and Yu in '92 and is known to admit an FPTAS [Bazgan, Santha, Tuza '98]. The best approximation schemes before this work had running time Opn 4 {εq [Melissinos, Pagourtzis '18], r Opn 2.3 {ε 2.6 q and r Opn 2 {ε 3 q [Alonistiotis et al. '22].In this work, we present an improved approximation scheme for Subset Sum Ratio running in time Opn{ε 0.9386 q. Here we assume that the items are given in sorted order, otherwise we need an additional running time of Opn log nq for sorting. Our improved running time simultaneously improves the dependence on n to linear and the dependence on 1{ε to sublinear.For comparison, the related Subset Sum problem admits an approximation scheme running in time Opn{εq [Gens, Levner '79]. If one would achieve an approximation scheme with running time r Opn{ε 0.99 q for Subset Sum, then one would falsify the Strong Exponential Time Hypothesis [Abboud, Bringmann, Hermelin, Shabtay '19] as well as the Min-Plus-Convolution Hypothesis [Bringmann, Nakos '21]. We thus establish that Subset Sum Ratio admits faster approximation schemes than Subset Sum. This comes as a surprise, since at any point in time before this work the best known approximation scheme for Subset Sum Ratio had a worse running time than the best known approximation scheme for Subset Sum.