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Removing all connections between two vertices s and z in a graph by removing a minimum number of vertices is a fundamental problem in algorithmic graph theory. This (s,z)-separation problem is well-known to be polynomial solvable and serves as an important primitive in many applications related to network connectivity. We study the NP-hard temporal (s,z)-separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle this problem by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general the problem remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of "change over time".

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of \emph{proportionality} is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the \emph{ordinal maximin share approximation}, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions. Appears in the 30th International Joint Conference on Artificial Intelligence (IJCAI), 2021

Large language models (LLMs), such as Codex, hold great promise in enhancing programming education by automatically generating feedback for students. We investigate using LLMs to generate feedback for fixing syntax errors in Python programs, a key scenario in introductory programming. More concretely, given a student's buggy program, our goal is to generate feedback comprising a fixed program along with a natural language explanation describing the errors/fixes, inspired by how a human tutor would give feedback. While using LLMs is promising, the critical challenge is to ensure high precision in the generated feedback, which is imperative before deploying such technology in classrooms. The main research question we study is: Can we develop LLMs-based feedback generation techniques with a tunable precision parameter, giving educators quality control over the feedback that students receive? To this end, we introduce PyFiXV, our technique to generate high-precision feedback powered by Codex. The key idea behind PyFiXV is to use a novel run-time validation mechanism to decide whether the generated feedback is suitable for sharing with the student; notably, this validation mechanism also provides a precision knob to educators. We perform an extensive evaluation using two real-world datasets of Python programs with syntax errors and show the efficacy of PyFiXV in generating high-precision feedback. Published in International Conference on Educational Data Mining (EDM) 2023

We present Variational Bayesian Network (VBN) - a novel Bayesian entity representation learning model that utilizes hierarchical and relational side information and is particularly useful for modeling entities in the ``long-tail'', where the data is scarce. VBN provides better modeling for long-tail entities via two complementary mechanisms: First, VBN employs informative hierarchical priors that enable information propagation between entities sharing common ancestors. Additionally, VBN models explicit relations between entities that enforce complementary structure and consistency, guiding the learned representations towards a more meaningful arrangement in space. Second, VBN represents entities by densities (rather than vectors), hence modeling uncertainty that plays a complementary role in coping with data scarcity. Finally, we propose a scalable Variational Bayes optimization algorithm that enables fast approximate Bayesian inference. We evaluate the effectiveness of VBN on linguistic, recommendations, and medical inference tasks. Our findings show that VBN outperforms other existing methods across multiple datasets, and especially in the long-tail.

We first propose a general method to construct the complete set of on-shell operator bases involving massive particles with any spins. To incorporate the non-abelian little groups of massive particles, the on-shell scattering amplitude basis should be factorized into two parts: one is charged, and the other one is neutral under little groups of massive particles. The complete set of these two parts can be systematically constructed by choosing some specific Young diagrams of Lorentz subgroup and global symmetry $U(N)$ respectively ($N$ is the number of external particles), without the equation of motion and integration by part redundancy. Thus the complete massive amplitude bases without any redundancies can be obtained by combining these two complete sets. Some examples are presented to explicitly demonstrate this method. This method is applicable for constructing amplitude bases involving identical particles, and all the bases can be constructed automatically by computer programs based on it. Comment: 12 pages

Chirality plays an important role in understanding the dynamics of quantum field theories. In this paper, we study the dynamics of models where renormalization group flows change the chiral structure of the theory. We introduce model building tools and construct models with a variety of chirality flows: from the appearance of new massless composite matter, to the development of mass gaps to completely general changes in the chiral matter content. The stability of chirally symmetric vacua is sensitive to the interplay between non-perturbative dynamics and deformations necessary to generate chirality flows. In particular, we show that chirality flows can be easily induced by deformations of s-confining models. On the other hand, in the absence of true s-confinement, the required deformations destabilize chirally symmetric ground states. 21 pages

Binary compositions of surface ligands are known to improve the colloidal stability and fluorescence quantum yield of nanocrystals (NCs), due to ligand-ligand interactions and surface organization. Herein, we follow the thermodynamics of a ligand exchange reaction of CdSe NCs with alkylthiols mixtures. The effects of ligand polarity and length difference on ligand packing were investigated using isothermal titration calorimetry (ITC). The thermodynamic signature of the formation of mixed ligand shells was observed. Correlating the experimental results with thermodynamic mixing models has allowed us to calculate the inter-chain interactions and infer the final ligand shell configuration. Our findings demonstrate that the small dimensions of the NCs and the subsequent increased interfacial region between dissimilar ligands, in contrast to macroscopic surfaces, allow the formation of a myriad of clustering patterns, controlled by the inter-ligand interactions. This work provides a fundamental understanding of the parameters determining the ligand shell structure and should help guide smart surface design toward NC-based applications. 28 pages, 4 figures; combines with SI file 24 pages, 14 figures, 9 tables

We argue that for any single-trace operator in ${\cal N}=4$ SYM theory there is a large twist double-scaling limit in which the Feynman graphs have an iterative structure. Such structure can be recast using a graph-building operator. Generically, this operator mixes between single-trace operators with different scaling limits. The mixing captures both the finite coupling spectrum and corrections away from the large twist limit. We first consider a class of short operators with gluons and fermions for which such mixing problems do not arise, and derive their finite coupling spectra. We then focus on a class of long operators with gluons that do mix. We invert their graph-building operator and prove its integrability. The picture that emerges from this work opens the door to a systematic expansion of ${\cal N}=4$ SYM theory around the large twist limit. 54 pages, 16 figures