Long monotone trails in random edge-labelings of random graphs

Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\emph{increasing} if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chv\'atal and Koml\'os raised the question of providing the worst-case estimates of the length of the longest increasing trail/path over all edge orderings of $K_n$. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is $n-1$, and the case of a path is still widely open. Recently Lavrov and Loh proposed to study the average case of this problem in which the edge ordering is chosen uniformly at random. They conjectured (and it was proved by Martinsson) that such an ordering with high probability (whp) contains an increasing Hamilton path. In this paper we consider random graph $G=G(n,p)$ and its edge ordering chosen uniformly at random. In this setting we determine whp the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average case of the result of Graham and Kleitman, showing that the random edge ordering of $K_n$ has whp an increasing trail of length $(1-o(1))en$ and this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erd\H{o}-Renyi graphs with $p=o(1)$

Truthful Mechanisms for Matching and Clustering in an Ordinal World

We study truthful mechanisms for matching and related problems in a partial information setting, where the agents' true utilities are hidden, and the algorithm only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph of agent utilities, but the algorithm can only elicit the agents' private information in the form of a preference ordering for each agent induced by the underlying weights. Against this backdrop, we design truthful algorithms to approximate the true optimum solution with respect to the hidden weights. Our techniques yield universally truthful algorithms for a number of graph problems: a 1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a 2-approximation algorithm for Max Traveling Salesman as long as the hidden weights constitute a metric. We also provide improved approximation algorithms for such problems when the agents are not able to lie about their preferences. Our results are the first non-trivial truthful approximation algorithms for these problems, and indicate that in many situations, we can design robust algorithms even when the agents may lie and only provide ordinal information instead of precise utilities.Comment: To appear in the Proceedings of WINE 201

Integrals of motion in the Many-Body localized phase

We construct a complete set of quasi-local integrals of motion for the many-body localized phase of interacting fermions in a disordered potential. The integrals of motion can be chosen to have binary spectrum $\{0,1\}$, thus constituting exact quasiparticle occupation number operators for the Fermi insulator. We map the problem onto a non-Hermitian hopping problem on a lattice in operator space. We show how the integrals of motion can be built, under certain approximations, as a convergent series in the interaction strength. An estimate of its radius of convergence is given, which also provides an estimate for the many-body localization-delocalization transition. Finally, we discuss how the properties of the operator expansion for the integrals of motion imply the presence or absence of a finite temperature transition.Comment: 65 pages, 12 figures. Corrected typos, added reference

Maximizing Neutrality in News Ordering

The detection of fake news has received increasing attention over the past few years, but there are more subtle ways of deceiving one's audience. In addition to the content of news stories, their presentation can also be made misleading or biased. In this work, we study the impact of the ordering of news stories on audience perception. We introduce the problems of detecting cherry-picked news orderings and maximizing neutrality in news orderings. We prove hardness results and present several algorithms for approximately solving these problems. Furthermore, we provide extensive experimental results and present evidence of potential cherry-picking in the real world.Comment: 14 pages, 13 figures, accepted to KDD '2