Billy Jin

Billy Jin, Nathan Klein, David P. Williamson

One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the Subtour LP relaxation of the TSP is equal to $\frac{4}{3}$. For 40 years, the best known upper bound was $1.5$, due to Wolsey. Recently, Karlin, Klein, and Oveis Gharan showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the maximum entropy algorithm is a $\frac{10}{7}$-approximation for half-integral cycle cut instances of the TSP. This class of instances contains examples which demonstrate the subtour LP has an integrality gap of at least $\frac{4}{3}$, as well as examples showing that the performance of the max entropy algorithm is no better than $\frac{11}{8}$. We note that in the authors recently gave an algorithm upper bounding the integrality gap of this class of instances by $\frac{4}{3}$, so this work does not (and could not) provide an improved bound on the integrality gap. However, since there is no reason to believe that the analysis of the maximum entropy algorithm on general instances is tight, our work provides hope (and potentially direction) for improved analysis on other instance classes.

Davin Choo, Billy Jin, Yongho Shin

Online bipartite matching is a fundamental problem in online optimization, extensively studied both in its integral and fractional forms due to its theoretical significance and practical applications, such as online advertising and resource allocation. Motivated by recent progress in learning-augmented algorithms, we study online bipartite fractional matching when the algorithm is given advice in the form of a suggested matching in each iteration. We develop algorithms for both the vertex-weighted and unweighted variants that provably dominate the naive "coin flip" strategy of randomly choosing between the advice-following and advice-free algorithms. Moreover, our algorithm for the vertex-weighted setting extends to the AdWords problem under the small bids assumption, yielding a significant improvement over the seminal work of Mahdian, Nazerzadeh, and Saberi (EC 2007, TALG 2012). Complementing our positive results, we establish a hardness bound on the robustness-consistency tradeoff that is attainable by any algorithm. We empirically validate our algorithms through experiments on synthetic and real-world data.