A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. This paper introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well.
-0.05in Locality issues have been thoroughly studied in the literature, via the analysis of various construction problems, including @math -coloring and Maximal Independent Set (MIS) @cite_39 @cite_40 @cite_24 @cite_35 @cite_20 @cite_0 @cite_19 , Minimum Spanning Tree (MST) @cite_31 @cite_26 @cite_33 , Maximal Matching @cite_38 , Maximum Weighted Matching @cite_8 @cite_23 @cite_9 , Minimum Dominating Set @cite_28 @cite_7 , Spanners @cite_4 @cite_32 @cite_12 , etc. For some problems (e.g., coloring @cite_40 @cite_24 @cite_19 ), there are still large gaps between the best known results on specific families of graphs (e.g., bounded degree graphs) and on arbitrary graphs. .0in
@cite_35 @cite_20 @cite_38 @cite_4 @cite_8 @cite_39 @cite_23 @cite_26 @cite_7 @cite_28 @cite_32 @cite_19 @cite_40 @cite_12 @cite_33 @cite_9 @cite_24 @cite_0 @cite_31
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The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). We deterministically compute a @D+1 coloring and a maximal independent set(MIS) in time O(@D^1^ ^2^+^@Q^(^1^ ^h^)+log^*n) for @D"1"+"[email protected][email protected]^1^+^i^ ^h, where @D"j is defined as the maximal number of nodes within distance j for a node and @D:[email protected]"1. Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(@D+log^*n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h>=36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O([email protected]+log^*n) for @[email protected][email protected](log^1^+^1^ ^l^o^g^^^*^nn) and @[email protected]?O(@D log^1^+^1^ ^l^o^g^^^*^nn). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest @D+1 coloring algorithm running in time O([email protected]+logn). The algorithm works without knowledge of @g and uses less than @D colors, i.e., (1-1 O(@g))@D with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number @g into account. We also improve on the state of the art deterministic computation of (2,c)-ruling sets. 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). Given a simple graph G=(V,E) and a set of sources S ⊆ V, denote for each node ν e V by Lν(∞) the lexicographically ordered list of distance source pairs (d(s,v),s), where s ∈ S. For integers d,k ∈ N∪ ∞ , we consider the source detection, or (S,d,k)-detection task, requiring each node v to learn the first k entries of Lν(∞) (if for all of them d(s,v) ≤ d) or all entries (d(s,v),s) ∈ Lν(∞) satisfying that d(s,v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k=∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d=∞ and S=V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d+k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions. 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 We analyze local search heuristics for the metric k-median and facility location problems. We define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution (obtained using this procedure) to the global optimum. For k-median, we show that local search with swaps has a locality gap of 5. Furthermore, if we permit up to p facilities to be swapped simultaneously, then the locality gap is 3+2 p. This is the first analysis of a local search for k-median that provides a bounded performance guarantee with only k medians. This also improves the previous known 4 approximation for this problem. For uncapacitated facility location, we show that local search, which permits adding, dropping, and swapping a facility, has a locality gap of 3. This improves the bound of 5 given by M. Korupolu, C. Plaxton, and R. Rajaraman [Analysis of a Local Search Heuristic for Facility Location Problems, Technical Report 98-30, DIMACS, 1998]. We also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility. For this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities. We prove that this local search has a locality gap between 3 and 4. 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). This paper studies the problem of constructing a minimum-weight spanning tree (MST) in a distributed network. This is one of the most important problems in the area of distributed computing. There is a long line of gradually improving protocols for this problem, and the state of the art today is a protocol with running time O(Λ(G)+n⋅log∗n) due to Kutten and Peleg [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27], where Λ(G) denotes the diameter of the graph G. Peleg and Rubinovich [D. Peleg, V. Rubinovich, A near-tight lower bound on the time complexity of distributed MST construction, in: Proc. 40th IEEE Symp. on Foundations of Computer Science, 1999, pp. 253–261] have shown that Ω˜(n) time is required for constructing MST even on graphs of small diameter, and claimed that their result “establishes the asymptotic near-optimality” of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. In this paper we refine this claim, and devise a protocol that constructs the MST in Ω˜(μ(G,ω)+n) rounds, where μ(G,ω) is the MST-radius of the graph. The ratio between the diameter and the MST-radius may be as large as Θ(n), and, consequently, on some inputs our protocol is faster than the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27] by a factor of Ω˜(n). Also, on every input, the running time of our protocol is never greater than twice the running time of the protocol of [S. Kutten, D. Peleg, Fast distributed construction of k-dominating sets and applications, J. Algorithms 28 (1998) 40–66; preliminary version appeared in: Proc. of 14th ACM Symp. on Principles of Distributed Computing, Ottawa, Canada, August 1995, pp. 20–27]. As part of our protocol for constructing an MST, we develop a protocol for constructing neighborhood covers with a drastically improved running time. The latter result may be of independent interest. We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called Nash-Williams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forests-decomposition. In this paper we study the problem of computing subgraphs of a certain configuration in a given topological graph G such that the number of crossings in the subgraph is minimum. The configurations that we consider are spanning trees, s-t paths, cycles, matchings, and @k-factors for @[email protected]? 1,2 . We show that it is NP-hard to approximate the minimum number of crossings for these configurations within a factor of k^1^-^@e for any @e>0, where k is the number of crossings in G. We then give a simple fixed-parameter algorithm that tests in O^@?(2^k) time whether G has a crossing-free configuration for any of the above, where the O^@?-notation neglects polynomial terms. For some configurations we have faster algorithms. The respective running times are O^@?(1.9999992^k) for spanning trees and O^@?((3)^k) for s-t paths and cycles. For spanning trees we also have an O^@?(1.968^k)-time Monte-Carlo algorithm. Each O^@?(@b^k)-time decision algorithm can be turned into an O^@?((@b+1)^k)-time optimization algorithm that computes a configuration with the minimum number of crossings. Abstract We propose a novel approach to solving the approximate k-nearest neighbor search problem in metric spaces. The search structure is based on a navigable small world graph with vertices corresponding to the stored elements, edges to links between them, and a variation of greedy algorithm for searching. The navigable small world is created simply by keeping old Delaunay graph approximation links produced at the start of construction. The approach is very universal, defined in terms of arbitrary metric spaces and at the same time it is very simple. The algorithm handles insertions in the same way as queries: by finding approximate neighbors for the inserted element and connecting it to them. Both search and insertion can be done in parallel requiring only local information from the structure. The structure can be made distributed. The accuracy of the probabilistic k-nearest neighbor queries can be adjusted without rebuilding the structure. The performed simulation for data in the Euclidean spaces shows that the structure built using the proposed algorithm has small world navigation properties with log 2 ( n ) insertion and search complexity at fixed accuracy, and performs well at high dimensionality. Simulation on a CoPHiR dataset revealed its high efficiency in case of large datasets (more than an order of magnitude less metric computations at fixed recall) compared to permutation indexes. Only 0.03 of the 10 million 208-dimensional vector dataset is needed to be evaluated to achieve 0.999 recall (virtually exact search). For recall 0.93 processing speed 2800 queries s can be achieved on a dual Intel X5675 Xenon server node with Java implementation. We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In this paper, we study unweighted δ-hyperbolic graphs. Using the Layering Partition technique, we show that every n-vertex δ-hyperbolic graph with δ≥1 2 has an additive O(δlog n)-spanner with at most O(δn) edges and provide a simpler, in our opinion, and faster construction of distance approximating trees of δ-hyperbolic graphs with an additive error O(δlog n). The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ≥1 2 admits a routing labeling scheme with O(δlog 2 n) bit labels, O(δlog n) additive stretch and O(log 2(4δ)) time routing protocol, and a distance labeling scheme with O(log 2 n) bit labels, O(δlog n) additive error and constant time distance decoder. In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as , in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5. We consider requests for capacity in a given tree network T e (V, E) where each edge e of the tree has some integer capacity ue. Each request f is a node pair with an integer demand df and a profit wf which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b-matching problems. When all demands are 1, we have the integer multicommodity flow problem. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand dmax is at most the minimum edge capacity umin, we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed. We study deterministic, distributed algorithms for two weak variants of the standard graph coloring problem. We consider defective colorings, i.e., colorings where nodes of a color class may induce a graph of maximum degree d for some parameter d>0. We also look at colorings where a minimum number of multi-chromatic edges is required. For an integer k>0, we call a coloring k-partially proper if every node v has at least min k,deg(v) neighbors with a different color. We show that for all d∈ 1,...,Δ , it is possible to compute a O(Δ2 d2)-coloring with defect d in time O(log*n) where Δ is the largest degree of the network graph. Similarly, for all k∈ 1,...,Δ , a k-partially proper O(k2)-coloring can be computed in O(log*n) rounds. As an application of our weak defective coloring algorithm, we obtain a faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees. We show that in time O(Δ+log*n), a (Δ+1)-coloring can be computed, a task for which the best previous algorithm required time O(Δ*log(Δ) + log*n). The same result holds for the problem of computing a maximal independent set. We define a natural notion of efficiency for approximate nearest-neighbor (ANN) search in general n-point metric spaces, namely the existence of a randomized algorithm which answers (1 + e)-ANN queries in polylog(n) time using only polynomial space. We then study which families of metric spaces admit efficient ANN schemes in the black-box model, where only oracle access to the distance function is given, and any query consistent with the triangle inequality may be asked.For e < 2 5, we offer a complete answer to this problem. Using the notion of metric dimension defined in [A. Gupta, et al, Bounded geometries, fractals, and low-distortion embeddings, in: 44th Annu. IEEE Symp. on Foundations of Computer Science, 2003, pp. 534-543] (a la [P. Assouad, Plongements lipschitziens dans Rn, Bull. Soc. Math. France 111 (4) (1983) 429-448]), we show that a metric space X admits an efficient (1+e)-ANN scheme for any e < 2 5 if and only if dim(X) = O(log log n). For coarser approximations, clearly the upper bound continues to hold, but there is a threshold at which our lower bound breaks down--this is precisely when points in the "ambient space" may begin to affect the complexity of "hard" subspaces S ⊆ X. Indeed, we give examples which show that dim(X) does not characterize the black-box complexity of ANN above the threshold.Our scheme for ANN in low-dimensional metric spaces is the first to yield efficient algorithms without relying on any additional assumptions on the input. In previous approaches (e.g., [K.L. Clarkson, Nearest neighbor queries in metric spaces, Discrete Comput. Geom. 22(1) (1999) 63-93; D. Karger, M. Ruhl, Finding nearest neighbors in growth-restricted metrics, in: 34th Annu. ACM Symp. on the Theory of Computing, 2002, pp. 63-66; R. Krauthgamer, J.R. Lee, Navigating nets: simple algorithms for proximity search, in: 15th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2004, pp. 791-801; K. Hildrum, et al, A note on finding nearest neighbors in growth-restricted metrics, in: Proc. of the 15th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2004, pp. 560-561]), even spaces with dim(X) = O(1) sometimes required Ω(n) query times. 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 . We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math .