Neural networks offer a promising data-driven approach to tackle computationally challenging optimization problems. In this work, we introduce neural approximation frameworks for a family of geometric "extent measure" problems, including shape-fitting descriptors (e.g. minimum enclosing ball or annulus). Central to our approach is the \textit{alignment} of our neural model with a new variant of the classical $\varepsilon$-kernel technique from computational geometry. In particular, we develop a new relaxed-$\varepsilon$-kernel theory that maintains the approximation guarantees of the classical $\varepsilon$-kernels but with the crucial benefit that it can be easily implemented with \textit{bounded model complexity} (i.e, constant number of parameters) by the simple SumFormer neural network. This leads to a simple neural model to approximate objects such as the directional width of any input point set, and empirically shows excellent out-of-distribution generalization. Many geometric extent measures, such as the minimum enclosing spherical shell, cannot be directly captured by $\varepsilon$-kernels. To this end, we show that an encode-process-decode framework with our kernel approximating NN used as the ``process'' module can approximate such extent measures, again, with bounded model complexity where parameters scale only with the approximation error $\varepsilon$ and not the size of the input set. Empirical results on diverse point‐cloud datasets demonstrate the practical performance of our models.

The Lightning Network (LN) is designed to improve the scalability of the Bitcoin blockchain by using off-chain payments 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 assess how network topology changes when participants start behaving rationally.

The security of the Lightning Network (LN) relies on the ability of the nodes to close a channel by settling their balances in case of malicious behaviors, 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 on layer-1, in the presence of a small coalition of adversarial nodes that forces a large set of honest users to interact with the blockchain, with the objective of locking or stealing users' funds. We show via simulations that, under historically-plausible congestion conditions, with mild statistical assumptions on channel balances, the proposed attacks can be performed by a small coalition.

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.

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 [20] and by Aumann and Rabani [5] that for general n-vertex graphs it is bounded by O(log n) and the Gupta-Newman-Rabinovich-Sinclair conjecture [13] 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 directed planar graphs is O(log3n). This is the first 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 [11] that the gap is Ω(n1/7), 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 [22] (with a slightly weaker bound). At the heart of our result we investigate low-distortion quasimetric embeddings into directed$e$1. More precisely, we construct O(log2n)-Lipschitz quasipartitions for the shortest-path quasimetric spaces of planar digrap$hs$, 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.

Karine Chubarian, Abdul Rafae Khan, Anastasios Sidiropoulos, Jia Xu. Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. 2021.

Recent years have witnessed a surge of approaches to use neural networks to help tackle combinatorial optimization problems, including graph optimization problems. However, theoretical understanding of such approaches remains limited. In this paper, we consider the geometric setting, where graphs are induced by points in a fixed dimensional Euclidean space. We show that several graph optimization problems can be approximated by an algorithm that is polynomial in graph size n via a framework we propose, call the Baker-paradigm. More importantly, a key advantage of the Baker-paradigm is that it decomposes the input problem into (at most linear number of) small sub-problems of fixed sizes (independent of the size of the input). For the family of such fixed-size sub-problems, we can now design neural networks with universal approximation guarantees to solve them. This leads to a mixed algorithmic-ML framework, which we call NN-Baker that has the capacity to approximately solve a family of graph optimization problems (e.g, maximum independent set and minimum vertex cover) in time linear to input graph size, and only polynomial to approximation parameter. We instantiate our NN-Baker by a CNN version and GNN version, and demonstrate the effectiveness and efficiency of our approach via a range of experiments.

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown in Sidiropoulos and Sridhar (33rd International Symposium on Computational Geometry (Brisbane 2017). Leibniz Int. Proc. Inform., vol. 77, # 58. Leibniz-Zent. Inform., Wadern, 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 integer $$d > 1$$ , any $$\delta \in (1,d)$$ , and any $$n \in {\mathbb {N}}$$ , there exists a set X of n points in $${\mathbb {R}}^{d}$$ , with fractal dimension $$\delta $$ such that for any $$\varepsilon > 0$$ and $$c \ge 1$$ , any c-spanner of X has treewidth $$\Omega ( n^{1-1/(\delta - \epsilon )}/c^{d-1} )$$ . This lower bound matches 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: The above results nearly match previously known upper bounds from [op. cit.], and generalize analogous lower bounds for the case of ambient dimension due to Marx and Sidiropoulos (30th Annual Symposium on Computational Geometry (Kyoto 2014), pp. 67–76. ACM, New York, 2014).

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in $\mathcal{T}$ is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in $\mathbb{R}^d$. We use these partitions with slack for embedding ultrametrics into $d$-dimensional Euclidean space: we give a $\mathop{\rm polylog}(Δ)$-approximation algorithm for embedding $n$-point ultrametrics into $\mathbb{R}^d$ with minimum distortion, where $Δ$ denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in $n$. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

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-ε)-fraction of all constraints, our algorithm computes a solution that satisfies (1-O(ε^{1/8}))-fraction of all constraints, in time O(n⁷) + (1/ε)^{O(1/ε^{1/8})} n.

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.

We study the problem of supervised learning a metric space under discriminative constraints. Given a universe X and sets S, D subset binom{X}{2} of similar and dissimilar pairs, we seek to find a mapping f:X -> 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 l. 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 quasi-polynomial time (QPTAS). We also present an exact algorithm for learning line metric spaces with perfect information in polynomial time. Our algorithms use a combination of tools from metric embeddings and graph partitioning, that could be of independent interest.

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 $\eps>0$, runs in time $\poly(n,d,1/\eps),$ 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 $\eps$ 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.

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 [15] and by Aumann and Rabani [3] that for general n-vertex graphs it is bounded by O(log n) and the Gupta-Newman-Rabinovich-Sinclair conjecture [9] asserts that it is O(1) for any family of graphs that excludes some fixed minor. The flow-cut gap is poorly understood for the case of directed graphs. We show that for uniform demands it is O(1) on directed series-parallel graphs, and on directed graphs of bounded pathwidth. These are the first constant upper bounds of this type for some non-trivial family of directed graphs. We also obtain O(1) upper bounds for the general multi-commodity flow-cut gap on directed trees and cycles. These bounds are obtained via new embeddings and Lipschitz quasipartitions for quasimetric spaces, which generalize analogous results form the metric case, and could be of independent interest. Finally, we discuss limitations of methods that were developed for undirected graphs, such as random partitions, and random embeddings.

Computing the Euler genus of a graph is a fundamental problem in algorithmic graph theory. It has been shown to be NP-hard by [Thomassen ’89, Thomassen ’97], even for cubic graphs, and a linear-time fixed-parameter algorithm has been obtained by [Mohar ’99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an O(1)-approximation is not ruled out, the currently best-known upper bound is a O(n1−α)-approximation, for some universal constant α>0 [Kawarabayashi and Sidiropoulos 2017].

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an $O(\sqrt{n})$-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved $\tilde{O}(n^{1/3})$ approximation for the case of metrics induced by unweighted trees.

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.

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, 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)* 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 poly(c); - an exact algorithm, which in time f'(H, c)* 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, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

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.

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by Thomassen [ J. Algorithms, 10 (1989), pp. 568--576; J. Combin. Theory, Ser. B, 57 (1993), pp. 196--206], and it is known to be fixed-parameter tractable. However, the approximability of the Euler genus is wide open. While the existence of an $O(1)$-approximation is not ruled out, only an $O(\sqrt{n})$-approximation [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317--322] is known even in bounded-degree graphs. In this paper we give a polynomial-time algorithm which, given a bounded-degree graph of Euler genus $g$, computes a drawing in a surface of Euler genus $g^{O(1)} \cdot \log^{O(1)} n$. Combined with the upper bound from [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317--322], our result also implies a $O(n^{1/2 - \alpha})$-approximation for some constant $\alpha>0$. Using our algorithm for approximating the Euler genus as a subroutine, we obtain, in a uniform fashion, algorithms with approximation ratios of the form $\mathsf{OPT}^{O(1)} \cdot \log^{O(1)} n$ for several related problems on bounded-degree graphs. These include the problems of orientable genus, crossing number, and planar edge and vertex deletion. Our algorithm and proof of correctness for the crossing number problem are simpler compared to the long and difficult proof in the recent breakthrough by Chuzhoy [ Proceedings of the ACM Symposium on Theory of Computing, 2011, pp. 303--312], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of the form $\operatorname{poly}(\mathsf{OPT},\log n)$. We also highlight some further applications of our results in the design of algorithms for graphs with small genus. Many such algorithms require that a drawing of the graph is given as part of the input. Our results imply that in several interesting cases, we can implement such algorithms even when the drawing is unknown.

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is NP-hard to approximate the GH distance better than a factor of 3 for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time O (min n , √ rn )-approximation algorithm for computing the GH distance between a pair of metric trees, where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an O (√ n )-approximation algorithm 1 .

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta). Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings. We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension. However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space.

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i.e.~static) clustering problems are not applicable anymore. We propose a set of optimization problems which we collectively refer to as 'temporal clustering'. The quality of a solution to a temporal clustering instance can be quantified using three parameters: the number of clusters $k$, the spatial clustering cost $r$, and the maximum cluster displacement $δ$ between consecutive time steps. We consider spatial clustering costs which generalize the well-studied $k$-center, discrete $k$-median, and discrete $k$-means objectives of classical clustering problems. We develop new algorithms that achieve trade-offs between the three objectives $k$, $r$, and $δ$. Our upper bounds are complemented by inapproximability results.

In the minimum planarization problem, given some n-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a logO(1)n-approximation algorithm for this problem on general graphs with running time nO(log n/log log n). We also obtain a O(nε)-approximation with running time nO(1/ε)for any arbitrarily small constant ε > 0. Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a nΩ(1)-approximation [1]. As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain O(n1/2+ε)approximation and n1/2 logO(1)n-approximation algorithms in time nO(1/ε)and nO(log n/log log n)respectively. The previously best-known result was a polynomial-time n9/10logO(1)n-approximation algorithm [2]. Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.

We study hierarchical clusterings of metric spaces that change over time. This is a natural geo- metric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric space, which naturally induces a hierarchical clustering of the set of points. We enforce temporal coherence among the embeddings by finding correspondences between successive pairs of ultrametric spaces which exhibit small distortion in the Gromov-Hausdorff sense. We present both upper and lower bounds on the approximability of the resulting optimization problems.

We initiate the study of metric embeddings with outliers. Given some finite metric space we wish to remove a small set of points and to find either an isometric or a low-distortion embedding of the remaining points into some host metric space. This is a natural problem that captures scenarios where a small fraction of points in the input corresponds to noise. We present polynomial-time approximation algorithms for computing outlier embeddings into Euclidean space, trees, and ultrametrics. In the case of isometric embeddings the objective is to minimize the number of outliers, while in the case of non-isometries we have a bi-criteria optimization problem where the goal is to minimize both the number of outliers and the distortion. We complement our approximation algorithms with NP-hardness results for these problems. We conclude with a brief experimental evaluation of our non-isometric outlier embedding on synthetic and real-world data sets.

We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}. We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by [Oveis Gharan and Saberi 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by [Erickson and Sidiropoulos 2014]. We show that for any class of \emph{nearly-embeddable} graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed $k\geq 0$, there exist $α, β>0$, such that ATSP on $n$-vertex $k$-nearly-embeddable graphs admits a $α$-approximation in time $O(n^β)$. The class of $k$-nearly-embeddable graphs contains graphs with at most $k$ apices, $k$ vortices of width at most $k$, and an underlying surface of either orientable or non-orientable genus at most $k$. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming. We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth $k$ (and hence on $k$-nearly embeddable graphs) requires time $n^{Ω(k)}$, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

Red-blue pebbling is a model of computation that captures the complexity of I/O operations in systems with external memory access. We focus on one-shot pebbling strategies, that is without re-computation. Prior work on this model has focused on finding upper and lower bounds on the I/O complexity of certain families of graphs. We give a polynomial-time bi-criteria approximation algorithm for this problem for graphs with bounded out-degree. More precisely, given a n-vertex DAG that admits a pebbling strategy with R red pebbles and I/O complexity opt, our algorithm outputs a strategy using O(R ⋅ log3/2 n) red pebbles, and I/O complexity O(opt ⋅ log3/2 n). We further extend our result to the generalization of red-blue pebble games that correspond to multi-level memory hierarchies. Finally, we complement our theoretical analysis with an experimental evaluation of our algorithm for red-blue pebbling.

Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by Thomassen [27] and a linear-time fixed-parameter algorithm has been obtained by Mohar [20]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of a constant factor approximation is not ruled out, the currently best-known upper bound is a trivial O(n/g)-approximation that follows from bounds on the Euler characteristic.

We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Given any algorithm that achieves an approximation ratio of f(n) on arbitrary n-vertex graphs as a black box, our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g. In particular, the O(log n/loglog n)-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log g/loglog g)-approximation algorithm for genus-g graphs. Moreover, recent results on approximating the genus of graphs imply that our O(log g/loglog g)-approximation algorithm can be applied to bounded-degree graphs even if no genus-g embedding of the graph is given. Our result improves and generalizes the o(√ g log g)-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g and requires a genus-g embedding as part of the input, even for bounded-degree graphs. Finally, our techniques yield a O(1)-approximation algorithm for ATSP on graphs of genus g with running time 2O(g) · nO(1).

We are studying d-dimensional geometric problems that have algorithms with 1−1/d appearing in the exponent of the running time, for example, in the form of 2n1−1/d or nk1−1/d. This means that these algorithms perform somewhat better in low dimensions, but the running time is almost the same for all large values d of the dimension. Our main result is showing that for some of these problems the dependence on 1−1/d is best possible under a standard complexity assumption. We show that, assuming the Exponential Time Hypothesis,

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by Thomassen [23], [24], and it is known to be fixed-parameter tractable. However, the approximability of the Euler genus is wide open. While the existence of an O(1)-approximation is not ruled out, only an O(√n)-approximation [3] is known even in bounded degree graphs. In this paper we give a polynomialtime algorithm which on input a bounded-degree graph of Euler genus g, computes a drawing into a surface of Euler genus gO(1)· logO(1)n. Combined with the upper bound from [3], our result also implies a O(n1/2-α)-approximation, for some constant α > 0. Using our algorithm for approximating the Euler genus as a subroutine, we obtain, in a unified fashion, algorithms with approximation ratios of the form OPTO(1)· logO(1)n for several related problems on bounded degree graphs. These include the problems of orientable genus, crossing number, and planar edge and vertex deletion problems. Our algorithm and proof of correctness for the crossing number problem is simpler compared to the long and difficult proof in the recent breakthrough by Chuzhoy [5], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of form poly(OPT, log n). We also highlight some further applications of our results in the design of algorithms for graphs with small genus. Many such algorithms require that a drawing of the graph is given as part of the input. Our results imply that in several interesting cases, we can implement such algorithms even when the drawing is unknown.

The planar embedding conjecture asserts that any planar metric admits an embedding into L_1 with constant distortion. This is a well-known open problem with important algorithmic implications, and has received a lot of attention over the past two decades. Despite significant efforts, it has been verified only for some very restricted cases, while the general problem remains elusive. In this paper we make progress towards resolving this conjecture. We show that every planar metric of non-positive curvature admits a constant-distortion embedding into L_1. This confirms the planar embedding conjecture for the case of non-positively curved metrics.

A classical result in metric geometry asserts that any n-point metric admits a linear-size spanner of dilation O(log n) [PS89]. More generally, for any c > 1, any metric space admits a spanner of size O(n1+1/c), and dilation at most c. This bound is tight assuming the well-known girth conjecture of Erdős [Erd63]. We show that for a metric induced by a set of n points in high-dimensional Euclidean space, it is possible to obtain improved dilation/size trade-offs. More specifically, we show that any n-point Euclidean metric admits a near-linear size spanner of dilation O(√log n). Using the LSH scheme of Andoni and Indyk [AI06] we further show that for any c > 1, there exist spanners of size roughly O(n1+1/c2) and dilation O(c). Finally, we also exhibit super-linear lower bounds on the size of spanners with constant dilation.

We describe a O(log n)-approximation algorithm for computing the homotopic Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms where known only for curves on the Euclidean plane with polygonal obstacles.

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log2 n). (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) · k · (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n · poly(d) · log3/2 n)-approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.

We exhibit an infinite doubling metric space (X,d) such that for any non-expansive f : X -> L1, there exists a pair x,y ∈ X with d(x,y) arbitrarily large, and such that |f(x)-f(y)\|1/d(x,y) ≲ √log log d(x,y)}/(log d(x,y)).

It has been shown by Indyk and Sidiropoulos that any graph of genus g > 0 can be stochastically embedded into a distribution over planar graphs with distortion 2O(g). This bound was later improved to O(g2) by Borradaile, Lee and Sidiropoulos. We give an embedding with distortion O(log g), which is asymptotically optimal. Apart from the improved distortion, another advantage of our embedding is that it can be computed in polynomial time. In contrast, the algorithm of requires solving an NP-hard problem. Our result implies in particular a reduction for a large class of geometric optimization problems from instances on genus-p graphs, to corresponding ones on planar graphs, with a O(log g) loss factor in the approximation guarantee.

We consider the problem of computing a minimum-distortion bijection between two point-sets in ℝ2. We prove the first non-trivial inapproximability result for this problem, for the case when the distortion is constant. More precisely, we show that there exist constants 0 < α < β, such that it is NP-hard to distinguish between spaces for which the distortion is either at most α, or at least β, under the Euclidean norm. This addresses a question of Kenyon, Rabani and Sinclair [KRS04], and extends a result due to Papadimitriou and Safra [PS05], who gave inapproximability for point-sets in ℝ3. We also apply similar ideas to the problem of computing a minimum-distortion embedding of a finite metric space into ℝ2. We obtain an analogous inapproximability result under the ℓ∞ norm for this problem. Inapproximability for the case of constant distortion was previously known only for dimension at least 3 [MS08].

We study the quantitative geometry of graphs in terms of their genus, using the structure of certain “cut graphs,” i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow/min-cut theorems, approximations for uniform and non-uniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. We show that such graphs admit an O(log g)-approximate multi-commodity max-flow/min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of , improving over the previous bound of [KLMN04]. We give an -approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kg D/n), improving over the previous bound of g2 · O(kg D/n) [KLPT09]. There is a lower bound of Ω(kg D/n), making this result almost tight. We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension . This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of . In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion , improving over the previous bound of . Our result is asymptotically optimal for every dependence g = g(n).

A common approach for dealing with large datasets is to stream over the input in one pass, and perform computations using sublinear resources. For truly massive datasets, however, even making a single pass over the data is prohibitive. Therefore, streaming computations must be distributed over many machines. In practice, obtaining significant speedups using distributed computation has numerous challenges including synchronization, load balancing, overcoming processor failures, and data distribution. Successful systems in practice such as Google's MapReduce and Apache's Hadoop address these problems by only allowing acertain classof highly distributable tasks defined by local computations that can be applied in any order to the input. The fundamental question that arises is: How does the class of computational tasks supported by these systems differ from the class for which streaming solutions exist? We introduce a simple algorithmic model for massive, unordered, distributed (mud) computation, as implemented by these systems. We show that in principle, mud algorithms are equivalent in power to symmetric streaming algorithms. More precisely, we show that any symmetric (order-invariant) function that can be computed by a streaming algorithm can also be computed by a mud algorithm, with comparable space and communication complexity. Our simulation uses Savitch's theorem and therefore has superpolynomial time complexity. We extend our simulation result to some natural classes of approximate and randomized streaming algorithms. We also give negative results, using communication complexity arguments to prove that extensions to private randomness, promise problems, and indeterminate functions are impossible. We also introduce an extension of the mud model to multiple keys and multiple rounds.

The relation between Datalog programs and homomorphism problems, and, between Datalog programs and bounded treewidth structures has been recognized for some time and given much attention recently. Additionally, the essential role of persistent variables (in program expansions) for solving several relevant problems has also started to be observed. In Afrati et al. [2005] the general notion of program persistencies was refined into four notions (two syntactical ones and two semantical ones) and the interrelationship between these four persistency numbers was studied. In the present article (1) we prove undecidability results concerning the semantical notions of persistency number--modulo equivalence, of persistency number and of characteristic integer, (2) we exhibit new classes of programs for which boundedness is undecidable and (3) we prove intractabiltity results concerning the syntactical notions of weak persistency number and of weak characteristic integer.

It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2O(g). By removing all $g$ handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].

We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the Gupta-Newman-Rabinovich-Sinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)-approximate multi-commodity max-flow/min-cut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest.

We introduce a hierarchical partitioning scheme of the Euclidean plane, called circular partitions. Such a partition consists of a hierarchy of convex polygons, each having small aspect ratio, and satisfying specified volume constraints. We apply these partitions to obtain a natural extension of the popular Treemap visualization method. Our proposed algorithm is not constrained in using only rectangles, and can achieve provably better guarantees on the aspect ratio of the constructed polygons.

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into R.d, where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n1/12is NP-hard. From this result we derive inapproximability with factor roughly n1/(22d-10)for every fixed d ges 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala). For d ges 3,we obtain a stronger inapproximability result by a different reduction: assuming PneNP, no polynomial- time algorithm can distinguish between spaces embeddable in R.d with constant distortion from spaces requiring distortion at least nc/d, for a constant c > 0. The exponent c/d has the correct order of magnitude, since every n-point metric space can be embedded in Rdwith distortion O(n2/dlog3/2n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in R2, while all not too large subspaces of it embed almost isometrically.

We introduce a new notion of embedding, called minimum-relaxation ordinal embedding , parallel to the standard notion of minimum-distortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much as possible. The (multiplicative) relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop several worst-case bounds and approximation algorithms on ordinal embedding. In particular, we establish that ordinal embedding has many qualitative differences from metric embedding, and we capture the ordinal behavior of ultrametrics and shortest-path metrics of unweighted trees.

A probabilistic C-embedding of a (guest) metric M into a collection of(host) metrics M'1, ..., M'k is a randomized mapping F of M intoone of the M'1, ..., M'k such that, for any two points p,q in theguest metric: The distance between F(p) and F(q) in any M'i is not smaller thanthe original distance between p and q. The expected distance between F(p) and F(q) in (random) M'i is notgreater than some constant C times the original distance, for C≥ 1. The constant C is called the distortion of the embedding. Low-distortion probabilistic embeddings enable reducing algorithmicproblems over "hard" guest metrics into "easy" host metrics.We show that every metric induced by a graph of bounded genus can beprobabilistically embedded into planar graphs, with constant distortion. The embedding can be computed efficiently, given a drawing of the graphon a genus-g surface.

We study the problem of minimum-distortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Low-distortion embeddings of ultrametrics into the plane help visualizing complex structures they often represent. Given an ultrametric, a natural question is whether we can efficiently find an optimal-distortion embedding of this ultrametric into the plane, and if not, whether we can design an efficient algorithm that produces embeddings with near-optimal distortion. We show that the problem of finding minimum-distortion embedding of ultrametrics into the plane is NP-hard, and thus approximation algorithms are called for. Given an input ultrametric M, let c denote the minimum distortion achievable by any embedding of M into the plane. Our main result is a linear-time algorithm that produces an O(c 3)-distortion embedding. This result can be generalized to embedding ultrametrics into ℜ d, for any d ≥ 2, with distortion c O(d), where c is the minimum distortion achievable for embedding the input ultrametric into ℜ d. Additionally, we show that any ultrametric can be embedded into the plane with distortion O ( √ n), and in general, into ℜ d with distortion d O(1) n 1/d. Combining the two results together, we obtain an O(n 1/3)-approximation algorithm for the problem of minimumdistortion embedding of ultrametrics into the plane.

A low-distortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Low-distortion embeddings have recently found numerous applications in computer science.Most of the known embedding results are "absolute",that is, of the form: any metric Y from a given class of metrics C can be embedded into a metric X with low distortion c. This is beneficial if one can guarantee low distortion for all metrics Y in C. However, in any situations, the worst-case distortion is too large to be meaningful. For example, if X is a line metric, then even very simple metrics (an n - point star or an n -point cycle) are embeddable into X only with distortion linear in n. Nevertheless, embeddings into the line (or into low-dimensional spaces) are important for many applications.A solution to this issue is to consider "relative" (or "approximation") embedding problems, where the goal is to design an (a-approxiation) algorithm which, given any metric X from C as an input, finds an embedding of X into Y which has distortion a *cY (X), where cY (X)is the best possible distortion of an embedding of X into Y.In this paper we show algorithms and hardness results for relative embedding problems.In particular we give: •an algorith that, given a general metric M, finds an embedding with distortion O (Δ3⁄4 poly(c line (M))), where Δ is the spread of M•an algorithm that,given a weighted tree etric M, finds an embedding with distortion poly(c line (M)) •a hardness result, showing that computing minimum line distortion is hard to approximate up to a factor polynomial in n,even for weighted tree metrics with spread Δ=n O (1).