In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted in many algorithmic applications. However, we still do not understand the interaction between various graph-theoretic (topological) properties of graphs, and the doubling (geometric) properties of the shortest-path metrics induced by them. For instance, the following natural question suggests itself: graph @math with @math such that the shortest path metric @math on @math is still doubling, and which agrees with @math on @math . This is often useful, given that unweighted graphs are often easier to reason about. We show that for any metric space @math , there is an graph @math with shortest-path metric @math such that -- for all @math , the distances @math , and -- the doubling dimension for @math is not much more than that of @math , where this change depends only on @math and not on the size of the graph. We show a similar result when both @math and @math are restricted to be trees: this gives a simpler proof that doubling trees embed into constant dimensional Euclidean space with constant distortion. We also show that our results are tight in terms of the tradeoff between distortion and dimension blowup.
Somewhat similar in spirit to our work is the @math -extension problem @cite_13 @cite_5 @cite_24 . Given a graph @math , the 0-extension ( cf. Lipschitz Extendability @cite_6 @cite_23 @cite_11 ) problem deals with extending a (Euclidean) embedding of the vertices of the graph to an embedding of the convex closure of the graph, while approximately preserving the Lipschitz constant of the embedding. Our results can be interpreted as analogues to the above where the goal is to approximately preserve the doubling dimension.
@cite_6 @cite_24 @cite_23 @cite_5 @cite_13 @cite_11
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There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^ d-1 joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs. Given a graph G = (V, E), a set of terminals T ⊆ V, and a metric D on T, the 0-extension problem is to assign vertices in V to terminals, so that the sum, over all edges e, of the distance (under D) between the terminals to which the end points of e are assigned, is minimized. This problem was first studied by Karzanov. Calinescu, Karloff and Rabani gave an O(logk) approximation algorithm based on a linear programming relaxation for the problem, where k is the number of terminals. We improve on this bound, and give an O(log k log log k) approximation algorithm for the problem. Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. 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 nonuniform 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 O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-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(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD 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 f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). 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 O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). We consider the problem of encoding graphs with n vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time in the @Q(logn)-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex. We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of n and m, to within lower order terms. We prove a lower bound in the cell probe model indicating it is impossible to achieve the information-theory lower bound up to lower order terms unless the graph is either too sparse (namely, m=o(n^@d) for any constant @d>0) or too dense (namely m=@w(n^2^-^@d) for any constant @d>0). Furthermore, we present a succinct encoding of graphs supporting aforementioned queries in constant time. The space requirement of the encoding is within a multiplicative 1+@e factor of the information-theory lower bound for any arbitrarily small constant @e>0. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly within lower order terms where the graph is very sparse (m=o(n^@d) for any constant @d>0), or very dense (m>n^2 lg^1^-^@dn for an arbitrarily small constant @d>0). We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where each vertex v is assigned a label from a set L of labels. We show how to construct a compact distance oracle that can answer queries of the form: "what is the distance from v to the closest lambda-labeled node" for a given node v in V and label lambda in L. This problem was introduced by Hermelin, Levy, Weimann and Yuster [ICALP 2011] where they present several results for this problem. In the first result, they show how to construct a vertex-label distance oracle of expected size O(kn^ 1+1 k ) with stretch (4k - 5) and query time O(k). In a second result, they show how to reduce the size of the data structure to O(kn ^ 1 k ) at the expense of a huge stretch, the stretch of this construction grows exponentially in k, (2^k-1). In the third result they present a dynamic vertex-label distance oracle that is capable of handling label changes in a sub-linear time. The stretch of this construction is also exponential in k, (2 3^ k-1 +1). We manage to significantly improve the stretch of their constructions, reducing the dependence on k from exponential to polynomial (4k-5), without requiring any tradeoff regarding any of the other variables. In addition, we introduce the notion of vertex-label spanners: subgraphs that preserve distances between every node v and label lambda. We present an efficient construction for vertex-label spanners with stretch-size tradeoff close to optimal. A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space Y⊇X, where the loss in the Lipschitz constant in the extension is independent of Y,Z, and f. We show that various classes of natural metric spaces are absolutely Lipschitz extendable. To cite this article: J.R. Lee, A. Naor, C. R. Acad. Sci. Paris, Ser. I 338 (2004). LetH=(T,U) be a connected graph,V?Ta set, andca non-negative function on the unordered pairs of elements ofV. In theminimum0-extension problem(*), one is asked to minimize the inner productcmover all metricsmonVsuch that (i)mcoincides with the distance function ofHwithinT; and (ii) eachv?Vis at zero distance from somes?T, i.e.m(v,s)=0. This problem is known to be NP-hard ifH=K3(as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable ifH=K2(the minimum cut problem) orH=K2,r(the minimum (2,r)-metric problem). We study problem (*) for all fixedH. More precisely, we consider the linear programming relaxation (**) of (*) that is obtained by dropping condition (ii) above, and callHminimizableif the minima in (*) and (**) coincide for allVandc. Note that for such anHproblem (*) is solvable in strongly polynomial time. Our main theorem asserts thatHis minimizable if and only ifHis bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense thatHcan be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics. Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in 4, we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphsH. Other results are also presented. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more). We describe algorithms for the problem of minimum distortion embeddings of finite metric spaces into the real line (or a finite subset of the line). The time complexities of our algorithms are parametrized by the values of the minimum distortion, a#x03B4;, and the spread, a#x0394;, of the point set we are embedding. We consider the problem of finding the minimum distortion bijection between two finite subsets of the Euclidean line. This problem was known to have an exact polynomial time solution when a#x03B4; is below a specific small constant, and hard to approximate within a factor of a#x03B4;1 -- a#x03B5;, when a#x03B4; is polynomially large. Let D be the largest adjacent pair distance, a value potentially much smaller than a#x0394;. Then we provide a a#x03B4;O(a#x03B4;2 log2 D) n O(1) time exact algorithm for this problem, which in particular yields a quasipolynomial running time for constant a#x03B4;, and polynomial D. For the more general problem of embedding any finite metric space (X, dX) into a finite subset of the line, Y, we provide a a#x03B4;O(a#x03B4;2) (mn)O(1) time O(1)-approximation algorithm (where |X|=n and |Y|=m), which runs in polynomial time provided a#x03B4; is a constant and a#x03B4; is polynomial. This in turn allows us to get a a#x03B4;O(a#x03B4;2) (n)O(1) time O(1)-approximation algorithm for embedding (X, dX) into the continuous real line. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . In this paper, we propose a novel technique for finding the graph embedding and function extension for directed graphs. We assume that the data points are sampled from a manifold and the similarity between the points is given by an asymmetric kernel. We provide a graph embedding algorithm which is motivated by Laplacian type operator on manifold. We also introduce a Nystrom type eigenfunctions extension which is used both for extending the embedding to new data points and to extend an empirical function on new data set. For extending the eigenfunctions to new points, we assume that only the distances of the new points from the labelled data are given. Simulation results demonstrate the performance of the proposed method in recovering the geometry of data and extending a function on new data points. HighlightsA novel technique for finding the graph embedding and function extension for directed graphs is proposed.The data points are assumed to be sampled from a manifold and the similarity between the points is given by an asymmetric kernel.A graph embedding algorithm motivated by Laplacian type operator on manifold is proposed.A Nystrom type eigenfunctions extension for extending the embedding an empirical function on new data set is proposed.In extension phase, we assume that only the distances of the new points to the labelled data are given. A Lipschitz embedding of a mctric space (X, d) into another one (Y, 8) is an application : X -»• Y such that : 3 1. B 6 ]0, + oo [. Vx. x ' e X, Ad(x, x ' ) 6( (x), f x' ) Bd(x, x'). We dcscribe here three mcthods to obtain Lipschitz embeddings of the metric space (R*, || ||) into some metric space (R", || || ). The third method allows us to minimize, for k = 1, the rank ofsuch an embedding (i.c. to obtain the minimal value of the integer n). We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. Given a Euclidean graph @math in @math with @math vertices and @math edges, we consider the problem of adding an edge to @math such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in @math @math time, resulting in a trivial @math @math -time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in @math @math time using @math @math space. To reduce the running time we consider several approximation algorithms. We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l2n, while (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3).It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space--that is defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n (λ2+1)), where λ2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0<δ≤½ is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quais-random graphs of logarithmic sphericity (Lemma 7).For the above bound to be linear, λ2 must be constant. We show that if the second eigenvalue of an n 2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2) entries (Theorem 5). Furthermore, for any 0<δ<½, and λ2, there are only finitely many δn-regular graphs with second eigenvalue at most λ2 (Corollary 4). Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. In the densest subgraph problem, given an edge-weighted undirected graph (G=(V,E,w) ), we are asked to find (S V ) that maximizes the density, i.e., w(S) |S|, where w(S) is the sum of weights of the edges in the subgraph induced by S. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of (S V ). Specifically, we introduce the f-density of (S V ), which is defined as w(S) f(|S|), where (f: Z _ 0 R _ 0 ) is a monotonically non-decreasing function. In the f-densest subgraph problem (f-DS), we aim to find (S V ) that maximizes the f-density w(S) f(|S|). Although f-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex concave size function f appropriately. For f-DS with convex function f, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for f-DS with concave function f, we propose an LP-based exact algorithm, a flow-based (O(|V|^3) )-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.