Thou Shalt Covet The Average Of Thy Neighbors' Cakes

We prove an $\Omega(n^2)$ lower bound on the query complexity of local proportionality in the Robertson-Webb cake-cutting model. Local proportionality requires that each agent prefer their allocation to the average of their neighbors' allocations in some undirected social network. It is a weaker fairness notion than envy-freeness, which also has query complexity $\Omega(n^2)$, and generally incomparable to proportionality, which has query complexity $\Theta(n \log n)$. This result separates the complexity of local proportionality from that of ordinary proportionality, confirming the intuition that finding a locally proportional allocation is a more difficult computational problem

Quasipolynomiality of the Smallest Missing Induced Subgraph

We study the problem of finding the smallest graph that does not occur as an induced subgraph of a given graph. This missing induced subgraph has at most logarithmic size and can be found by a brute-force search, in an $n$-vertex graph, in time $n^{O(\log n)}$. We show that under the Exponential Time Hypothesis this quasipolynomial time bound is optimal. We also consider variations of the problem in which either the missing subgraph or the given graph comes from a restricted graph family; for instance, we prove that the smallest missing planar induced subgraph of a given planar graph can be found in polynomial time.Comment: 10 pages, 1 figure. To appear in J. Graph Algorithms Appl. This version updates an author affiliatio

On the Complexity of BWT-Runs Minimization via Alphabet Reordering

The Burrows-Wheeler Transform (BWT) has been an essential tool in text compression and indexing. First introduced in 1994, it went on to provide the backbone for the first encoding of the classic suffix tree data structure in space close to the entropy-based lower bound. Recently, there has been the development of compact suffix trees in space proportional to "$r$", the number of runs in the BWT, as well as the appearance of $r$ in the time complexity of new algorithms. Unlike other popular measures of compression, the parameter $r$ is sensitive to the lexicographic ordering given to the text's alphabet. Despite several past attempts to exploit this, a provably efficient algorithm for finding, or approximating, an alphabet ordering which minimizes $r$ has been open for years. We present the first set of results on the computational complexity of minimizing BWT-runs via alphabet reordering. We prove that the decision version of this problem is NP-complete and cannot be solved in time $2^{o(\sigma + \sqrt{n})}$ unless the Exponential Time Hypothesis fails, where $\sigma$ is the size of the alphabet and $n$ is the length of the text. We also show that the optimization problem is APX-hard. In doing so, we relate two previously disparate topics: the optimal traveling salesperson path and the number of runs in the BWT of a text, providing a surprising connection between problems on graphs and text compression. Also, by relating recent results in the field of dictionary compression, we illustrate that an arbitrary alphabet ordering provides a $O(\log^2 n)$-approximation. We provide an optimal linear-time algorithm for the problem of finding a run minimizing ordering on a subset of symbols (occurring only once) under ordering constraints, and prove a generalization of this problem to a class of graphs with BWT like properties called Wheeler graphs is NP-complete

NormBank: A Knowledge Bank of Situational Social Norms

We present NormBank, a knowledge bank of 155k situational norms. This resource is designed to ground flexible normative reasoning for interactive, assistive, and collaborative AI systems. Unlike prior commonsense resources, NormBank grounds each inference within a multivalent sociocultural frame, which includes the setting (e.g., restaurant), the agents' contingent roles (waiter, customer), their attributes (age, gender), and other physical, social, and cultural constraints (e.g., the temperature or the country of operation). In total, NormBank contains 63k unique constraints from a taxonomy that we introduce and iteratively refine here. Constraints then apply in different combinations to frame social norms. Under these manipulations, norms are non-monotonic - one can cancel an inference by updating its frame even slightly. Still, we find evidence that neural models can help reliably extend the scope and coverage of NormBank. We further demonstrate the utility of this resource with a series of transfer experiments

Light Euclidean Spanners with Steiner Points

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any $d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}{\epsilon}$), and also shows the existence of point sets in $\mathbb{R}^d$ for which any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon^{-d})$. Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with lightness $O(\epsilon^{-1} \log \Delta)$, where $\Delta$ is the spread of the point set. In the regime of $\Delta \ll 2^{1/\epsilon}$, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications $\epsilon$ often controls the precision, and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for spread polynomially bounded in $1/\epsilon$, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in $O_{\epsilon}(n)$ time for polynomially bounded spread, where $O_{\epsilon}$ hides a factor of $\mathrm{poly}(\frac{1}{\epsilon})$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(\epsilon^{-(d+1)/2} + \epsilon^{-2}\log \Delta)$ for any $3\leq d = O(1)$ and any $\epsilon = \Omega(n^{-\frac{1}{d-1}})$.Comment: 23 pages, 2 figures, to appear in ESA 2

On Constructing Spanners from Random Gaussian Projections

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA\u2712) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA\u2721), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work

What Do Our Choices Say About Our Preferences?

Taking online decisions is a part of everyday life. Think of buying a house, parking a car or taking part in an auction. We often take those decisions publicly, which may breach our privacy - a party observing our choices may learn a lot about our preferences. In this paper we investigate the online stopping algorithms from the privacy preserving perspective, using a mathematically rigorous differential privacy notion. In differentially private algorithms there is usually an issue of balancing the privacy and utility. In this regime, in most cases, having both optimality and high level of privacy at the same time is impossible. We propose a natural mechanism to achieve a controllable trade-off, quantified by a parameter, between the accuracy of the online algorithm and its privacy. Depending on the parameter, our mechanism can be optimal with weaker differential privacy or suboptimal, yet more privacy-preserving. We conduct a detailed accuracy and privacy analysis of our mechanism applied to the optimal algorithm for the classical secretary problem. Thereby the classical notions from two distinct areas - optimal stopping and differential privacy - meet for the first time.Comment: 22 pages, 6 figure