arXiv subjects

Anastasios Sidiropoulos

Publications and source records attributed to Anastasios Sidiropoulos.

At least 19 recordsLinked to original sources

arXiv2023

Rational Economic Behaviours in the Bitcoin Lightning Network

The Bitcoin Lightning Network (LN) is designed to improve the scalability of blockchain systems by using off-chain payment paths to settle transactions in a faster, cheaper, and more private manner. This work aims to empirically study LN's fee revenue for network participants. Under realistic assumptions on payment amounts, routing algorithms and traffic distribution, we analyze the economic returns of the network's largest routing nodes which currently hold the network together, and assess whether the centralizing tendency is incentive-compatible from an economic viewpoint. Moreover, since recent literature has proved that participation is economically irrational for the majority of large nodes, we evaluate the long-term impact on the network topology when participants start behaving rationally.

cs.GT↗

arXiv2023

Geometric Algorithms for $k$-NN Poisoning

We propose a label poisoning attack on geometric data sets against $k$-nearest neighbor classification. We provide an algorithm that can compute an $\varepsilon n$-additive approximation of the optimal poisoning in $n\cdot 2^{2^{O(d+k/\varepsilon)}}$ time for a given data set $X \in \mathbb{R}^d$, where $|X| = n$. Our algorithm achieves its objectives through the application of multi-scale random partitions.

cs.LG↗

arXiv2022

Mass Exit Attacks on the Lightning Network

The Lightning Network (LN) has enjoyed rapid growth over recent years, and has become the most popular scaling solution for the Bitcoin blockchain. The security of the LN relies on the ability of the nodes to close a channel by settling their balances, which requires confirming a transaction on the Bitcoin blockchain within a pre-agreed time period. We study the susceptibility of the LN to mass exit attacks in case of high transaction congestion, in the presence of a small coalition of adversarial nodes that forces a large set of honest users to interact with the blockchain. We focus on two types of attacks: (i) The first is a zombie attack, where a set of k nodes become unresponsive with the goal of locking the funds of many channels for a period of time longer than what the LN protocol dictates. (ii) The second is a mass double-spend attack, where a set of k nodes attempt to steal funds by submitting many closing transactions that settle channels using expired protocol states; this causes many honest nodes to have to quickly respond by submitting invalidating transactions. We show via simulations that, under historically plausible congestion conditions, with mild statistical assumptions on channel balances, both attacks can be performed by a very small coalition. To perform our simulations, we formulate the problem of finding a worst-case coalition of k adversarial nodes as a graph cut problem. Our experimental findings are supported by theoretical justifications based on the scale-free topology of the LN.

cs.CR↗

arXiv2021

Embeddings of Planar Quasimetrics into Directed \ell_1$ and Polylogarithmic Approximation for Directed Sparsest-Cut

The multi-commodity flow-cut gap is a fundamental parameter that affects the performance of several divide \& conquer algorithms, and has been extensively studied for various classes of undirected graphs. It has been shown by Linial, London and Rabinovich and by Aumann and Rabani that for general $n$-vertex graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture asserts that it is $O(1)$ for any family of graphs that excludes some fixed minor. We show that the multicommodity flow-cut gap on \emph{directed} planar graphs is $O(\log^3 n)$. This is the first \emph{sub-polynomial} bound for any family of directed graphs of super-constant treewidth. We remark that for general directed graphs, it has been shown by Chuzhoy and Khanna that the gap is $\widetilde{\Omega}(n^{1/7})$, even for directed acyclic graphs. As a direct consequence of our result, we also obtain the first polynomial-time polylogarithmic-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut, and the Directed Multicut problems for directed planar graphs, which extends the long-standing result for undirectd planar graphs by Rao (with a slightly weaker bound). At the heart of our result we investigate low-distortion quasimetric embeddings into \emph{directed} $\ell_1$. More precisely, we construct $O(\log^2 n)$-Lipschitz quasipartitions for the shortest-path quasimetric spaces of planar digraphs, which generalize the notion of Lipschitz partitions from the theory of metric embeddings. This construction combines ideas from the theory of bi-Lipschitz embeddings, with tools form data structures on directed planar graphs.

cs.DS↗

arXiv2020

Maximizing coverage while ensuring fairness: a tale of conflicting objective

Ensuring fairness in computational problems has emerged as a $key$ topic during recent years, buoyed by considerations for equitable resource distributions and social justice. It $is$ possible to incorporate fairness in computational problems from several perspectives, such as using optimization, game-theoretic or machine learning frameworks. In this paper we address the problem of incorporation of fairness from a $combinatorial$ $optimization$ perspective. We formulate a combinatorial optimization framework, suitable for analysis by researchers in approximation algorithms and related areas, that incorporates fairness in maximum coverage problems as an interplay between $two$ conflicting objectives. Fairness is imposed in coverage by using coloring constraints that $minimizes$ the discrepancies between number of elements of different colors covered by selected sets; this is in contrast to the usual discrepancy minimization problems studied extensively in the literature where (usually two) colors are $not$ given $a$ $priori$ but need to be selected to minimize the maximum color discrepancy of $each$ individual set. Our main results are a set of randomized and deterministic approximation algorithms that attempts to $simultaneously$ approximate both fairness and coverage in this framework.

cs.CC↗

arXiv2020

Learning Lines with Ordinal Constraints

We study the problem of finding a mapping $f$ from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points $(u,v,w)$ asserts that $|f(u)-f(v)|<|f(u)-f(w)|$. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies $(1-\varepsilon)$-fraction of all constraints, our algorithm computes a solution that satisfies $(1-O(\varepsilon^{1/8}))$-fraction of all constraints, in time $O(n^7) + (1/\varepsilon)^{O(1/\varepsilon^{1/8})} n$.

cs.DS↗

arXiv2020

Computing Bi-Lipschitz Outlier Embeddings into the Line

The problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion $O(c^2)$, where $c$ denotes the optimal distortion [B\u{a}doiu \etal~2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of \emph{outliers}. Specifically, we say that a metric space $(X,\rho)$ admits a $(k,c)$-embedding if there exists $K\subset X$, with $|K|=k$, such that $(X\setminus K, \rho)$ admits an embedding into the line with distortion at most $c$. Given $k\geq 0$, and a metric space that admits a $(k,c)$-embedding, for some $c\geq 1$, our algorithm computes a $({\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(k, c, \log n), {\mathsf p}{\mathsf o}{\mathsf l}{\mathsf y}(c))$-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion.

cs.DS↗

arXiv2019

A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families

We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given $n$ points in $\mathbb{R}^d$ and an accuracy parameter $\epsilon>0$, runs in time $poly(n,d,1/\epsilon),$ and returns a log-concave distribution which, with high probability, has the property that the likelihood of the $n$ points under the returned distribution is at most an additive $\epsilon$ less than the maximum likelihood that could be achieved via any log-concave distribution. This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task. Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, "locally" possesses key properties of exponential families. This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method. Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.

cs.DS↗

arXiv2019

Robust Mahalanobis Metric Learning via Geometric Approximation Algorithms

Learning Mahalanobis metric spaces is an important problem that has found numerous applications. Several algorithms have been designed for this problem, including Information Theoretic Metric Learning (ITML) [Davis et al. 2007] and Large Margin Nearest Neighbor (LMNN) classification [Weinberger and Saul 2009]. We study the problem of learning a Mahalanobis metric space in the presence of adversarial label noise. To that end, we consider a formulation of Mahalanobis metric learning as an optimization problem, where the objective is to minimize the number of violated similarity/dissimilarity constraints. We show that for any fixed ambient dimension, there exists a fully polynomial-time approximation scheme (FPTAS) with nearly-linear running time. This result is obtained using tools from the theory of linear programming in low dimensions. As a consequence, we obtain a fully-parallelizable algorithm that recovers a nearly-optimal metric space, even when a small fraction of the labels is corrupted adversarially. We also discuss improvements of the algorithm in practice, and present experimental results on real-world, synthetic, and poisoned data sets.

cs.LG↗

arXiv2018

A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities

We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities. Our main result is the first computationally efficient algorithm for this problem. In more detail, we give an algorithm that, on input a set of $n$ points in $\mathbb{R}^d$ and an accuracy parameter $\epsilon>0$, it runs in time $\text{poly}(n, d, 1/\epsilon)$, and outputs a log-concave density that with high probability maximizes the log-likelihood up to an additive $\epsilon$. Our approach relies on a natural convex optimization formulation of the underlying problem that can be efficiently solved by a projected stochastic subgradient method. The main challenge lies in showing that a stochastic subgradient of our objective function can be efficiently approximated. To achieve this, we rely on structural results on approximation of log-concave densities and leverage classical algorithmic tools on volume approximation of convex bodies and uniform sampling from convex sets.

cs.DS↗

arXiv2018

Chasing Similarity: Distribution-aware Aggregation Scheduling (Extended Version)

Parallel aggregation is a ubiquitous operation in data analytics that is expressed as GROUP BY in SQL, reduce in Hadoop, or segment in TensorFlow. Parallel aggregation starts with an optional local pre-aggregation step and then repartitions the intermediate result across the network. While local pre-aggregation works well for low-cardinality aggregations, the network communication cost remains significant for high-cardinality aggregations even after local pre-aggregation. The problem is that the repartition-based algorithm for high-cardinality aggregation does not fully utilize the network. In this work, we first formulate a mathematical model that captures the performance of parallel aggregation. We prove that finding optimal aggregation plans from a known data distribution is NP-hard, assuming the Small Set Expansion conjecture. We propose GRASP, a GReedy Aggregation Scheduling Protocol that decomposes parallel aggregation into phases. GRASP is distribution-aware as it aggregates the most similar partitions in each phase to reduce the transmitted data size in subsequent phases. In addition, GRASP takes the available network bandwidth into account when scheduling aggregations in each phase to maximize network utilization. The experimental evaluation on real data shows that GRASP outperforms repartition-based aggregation by 3.5x and LOOM by 2.0x.

cs.DB↗

arXiv2018

Algorithms for metric learning via contrastive embeddings

We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into some target metric space $M=(Y,\rho)$, such that similar objects are mapped to points at distance at most $u$, and dissimilar objects are mapped to points at distance at least $\ell$. More generally, the goal is to find a mapping of maximum accuracy (that is, fraction of correctly classified pairs). We propose approximation algorithms for various versions of this problem, for the cases of Euclidean and tree metric spaces. For both of these target spaces, we obtain fully polynomial-time approximation schemes (FPTAS) for the case of perfect information. In the presence of imperfect information we present approximation algorithms that run in quasipolynomial time (QPTAS). Our algorithms use a combination of tools from metric embeddings and graph partitioning, that could be of independent interest.

cs.CG↗

arXiv2018

Alleviating partisan gerrymandering: can math and computers help to eliminate wasted votes?

Partisan gerrymandering is a major cause for voter disenfranchisement in United States. However, convincing US courts to adopt specific measures to quantify gerrymandering has been of limited success to date. Recently, Stephanopoulos and McGhee introduced a new and precise measure of partisan gerrymandering via the so-called "efficiency gap" that computes the absolutes difference of wasted votes between two political parties in a two-party system. Quite importantly from a legal point of view, this measure was found legally convincing enough in a US appeals court in a case that claims that the legislative map of the state of Wisconsin was gerrymandered; the case is now pending in US Supreme Court. In this article, we show the following: (a) We provide interesting mathematical and computational complexity properties of the problem of minimizing the efficiency gap measure. To the best of our knowledge, these are the first non-trivial theoretical and algorithmic analyses of this measure of gerrymandering. (b) We provide a simple and fast algorithm that can "un-gerrymander" the district maps for the states of Texas, Virginia, Wisconsin and Pennsylvania by bring their efficiency gaps to acceptable levels from the current unacceptable levels. Our work thus shows that, notwithstanding the general worst case approximation hardness of the efficiency gap measure as shown by us,finding district maps with acceptable levels of efficiency gaps is a computationally tractable problem from a practical point of view. Based on these empirical results, we also provide some interesting insights into three practical issues related the efficiency gap measure. We believe that, should the US Supreme Court uphold the decision of lower courts, our research work and software will provide a crucial supporting hand to remove partisan gerrymandering.

cs.CY↗

arXiv2018

Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities

We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\mathbb{R}^d$, for all $d \geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions. In more detail, we prove that for any $d \geq 1$ and $\epsilon>0$, given $\tilde{O}_d((1/\epsilon)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\mathbb{R}^d$, the MLE outputs a hypothesis $h$ that with high probability is $\epsilon$-close to $f_0$, in squared Hellinger loss. A sample complexity lower bound of $\Omega_d((1/\epsilon)^{(d+1)/2})$ was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an $\tilde{O}(1/\epsilon)$ factor.

math.ST↗

arXiv2018

Sketching and Clustering Metric Measure Spaces

Two important optimization problems in the analysis of geometric data sets are clustering and sketching. Here, clustering refers to the problem of partitioning some input metric measure space (mm-space) into k clusters, minimizing some objective function f. Sketching, on the other hand, is the problem of approximating some mm-space by a smaller one supported on a set of k points. Specifically, we define the k-sketch of some mm-space M to be the nearest neighbor of M in the set of k-point mm-spaces, under some distance function \rho on the set of mm-spaces. In this paper, we demonstrate a duality between general classes of clustering and sketching problems. We present a general method for efficiently transforming a solution for a clustering problem to a solution for a sketching problem, and vice versa, with approximately equal cost. More specifically, we obtain the following results. 1. For metric spaces, we consider the case where the clustering objective is minimizing the maximum cluster diameter. We show that the ratio between the sketching and clustering objectives is constant over compact metric spaces. 2. We extend these results to the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the l_p-diameters} of the clusters. 3. We consider two competing notions of sketching for mm-spaces, with one of them being more demanding than the other. These notions arise from two different definitions of p-Gromov-Wasserstein distance that have appeared in the literature. We then prove that whereas the gap between these can be arbitrarily large, in the case of doubling metric spaces the resulting sketching objectives are polynomially related.

cs.CG↗

arXiv2017

Algorithms for low-distortion embeddings into arbitrary 1-dimensional spaces

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric $X$. Computing such an embedding (exactly or approximately) is a non-trivial task even when $X$ is the metric induced by a path, or, equivalently, into the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph $H$, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs $G$, $H$ and integer $c$, is it possible to embed $G$ with distortion $c$ into a graph homeomorphic to $H$? Then embedding into the line is the special case $H=K_2$, and embedding into the cycle is the case $H=K_3$, where $K_k$ denotes the complete graph on $k$ vertices. For this problem we give -an approximation algorithm, which in time $f(H)\cdot \text{poly} (n)$, for some function $f$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $\text{poly}(c)$; -an exact algorithm, which in time $f'(H, c)\cdot \text{poly} (n)$, for some function $f'$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $c$. Prior to our work, $\text{poly}(\mathsf{OPT})$-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

cs.CG↗

arXiv2017

Fractal dimension and lower bounds for geometric problems

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension. More specifically, we show that for any set of $n$ points in $d$-dimensional Euclidean space, of fractal dimension $\delta\in (1,d)$, for any $\epsilon >0$ and $c\geq 1$, any $c$-spanner must have treewidth at least $\Omega \left( \frac{n^{1-1/(\delta - \epsilon)}}{c^{d-1}} \right)$, matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type. For any $\delta \in (1,d)$ and any $\epsilon >0$ we show that: 1) $d$-dimensional Euclidean TSP on $n$ points with fractal dimension at most $\delta$ cannot be solved in time $2^{O\left(n^{1-1/(\delta - \epsilon)} \right)}$. The best-known upper bound is $2^{O(n^{1-1/\delta} \log n)}$. 2) The problem of finding $k$-pairwise non-intersecting $d$-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most $\delta$ cannot be solved in time $f(k)n^{O \left(k^{1-1/(\delta - \epsilon)}\right)}$ for any computable function $f$. The best-known upper bound is $n^{O(k^{1-1/\delta} \log n)}$. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014].

cs.CC↗

arXiv2017

Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands.

cs.DS↗