arXiv subjects

Michael Dinitz

Publications and source records attributed to Michael Dinitz.

At least 55 records · Page 3Linked to original sources

arXiv2019

The Norms of Graph Spanners

A $t$-spanner of a graph $G$ is a subgraph $H$ in which all distances are preserved up to a multiplicative $t$ factor. A classical result of Althöfer et al. is that for every integer $k$ and every graph $G$, there is a $(2k-1)$-spanner of $G$ with at most $O(n^{1+1/k})$ edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "few" nodes of "large" degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the $\ell_p$-norm of their degree vector, thus simultaneously modeling the number of edges (the $\ell_1$-norm) and the maximum degree (the $\ell_{\infty}$-norm). We give precise upper bounds for all ranges of $p$ and stretch $t$: we prove that the greedy $(2k-1)$-spanner has $\ell_p$ norm of at most $\max(O(n), O(n^{(k+p)/(kp)}))$, and that this bound is tight (assuming the Erdős girth conjecture). We also study universal lower bounds, allowing us to give "generic" guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the $\ell_1$ and $\ell_{\infty}$ norm. Finally, we show that at least in some situations, the $\ell_p$ norm behaves fundamentally differently from $\ell_1$ or $\ell_{\infty}$: there are regimes ($p=2$ and stretch $3$ in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.

cs.DS↗

arXiv2018

Distributed Algorithms for Minimum Degree Spanning Trees

The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $\hat d = O(d\log{n})$. It requires $O((D + \sqrt{n}) \log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexity for the related minimum spanning tree problem. Our second result refines this approximation factor by constructing a tree with maximum degree $\hat d = O(d + \log{n})$, though at the cost of additional polylogarithmic factors in the round complexity. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990's, our results are first efficient distributed solutions for this problem.

cs.DS↗

arXiv2018

Characterizing Demand Graphs for (Fixed-Parameter) Shallow-Light Steiner Network

We consider the Shallow-Light Steiner Network problem from a fixed-parameter perspective. Given a graph $G$, a distance bound $L$, and $p$ pairs of vertices $(s_1,t_1),\cdots,(s_p,t_p)$, the objective is to find a minimum-cost subgraph $G'$ such that $s_i$ and $t_i$ have distance at most $L$ in $G'$ (for every $i \in [p]$). Our main result is on the fixed-parameter tractability of this problem with parameter $p$. We exactly characterize the demand structures that make the problem "easy", and give FPT algorithms for those cases. In all other cases, we show that the problem is W$[1]$-hard. We also extend our results to handle general edge lengths and costs, precisely characterizing which demands allow for good FPT approximation algorithms and which demands remain W$[1]$-hard even to approximate.

cs.DS↗

arXiv2017

Large Fixed-Diameter Graphs are Good Expanders

We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that "sufficiently large" graphs of fixed diameter and degree must be "good" expanders. We prove this statement for various definitions of "sufficiently large" (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research.

math.CO↗

arXiv2017

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $Ω(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.

cs.DS↗

arXiv2017

Distributed Distance-Bounded Network Design Through Distributed Convex Programming

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al.~2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call "distance-bounded network design convex programs". These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in $O( (D/ε) \log n)$ rounds in the $\mathcal{LOCAL}$ model and finds a $(1+ε)$-approximation to the optimal LP solution for any $0 < ε\leq 1$, where $D$ is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic $3$- and $4$-Spanner, Directed $k$-Spanner, Lowest Degree $k$-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any "heavy" computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

cs.DS↗

arXiv2016

Approximating Approximate Distance Oracles

Given a finite metric space $(V,d)$, an approximate distance oracle is a data structure which, when queried on two points $u,v \in V$, returns an approximation to the the actual distance between $u$ and $v$ which is within some bounded stretch factor of the true distance. There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization. If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input? Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes. In particular, we give an $O(\log n)$-approximation to the problem of finding the smallest stretch $3$ Thorup-Zwick distance oracle, as well as the problem of finding the smallest Pǎtraşcu-Roditty distance oracle. We also prove a matching $Ω(\log n)$ lower bound for both problems, and an $Ω(n^{\frac{1}{k}-\frac{1}{2^{k-1}}})$ integrality gap for the more general stretch $(2k-1)$ Thorup-Zwick distance oracle. We also consider the problem of approximating the best TZ or PR approximate distance oracle \emph{with outliers}, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers.

cs.DS↗

arXiv2016

Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion

In the Minimum k-Union problem (MkU) we are given a set system with n sets and are asked to select k sets in order to minimize the size of their union. Despite being a very natural problem, it has received surprisingly little attention: the only known approximation algorithm is an $O(\sqrt{n})$-approximation due to [Chlamtáč et al APPROX '16]. This problem can also be viewed as the bipartite version of the Small Set Vertex Expansion problem (SSVE), which we call the Small Set Bipartite Vertex Expansion problem (SSBVE). SSVE, in which we are asked to find a set of k nodes to minimize their vertex expansion, has not been as well studied as its edge-based counterpart Small Set Expansion (SSE), but has recently received significant attention, e.g. [Louis-Makarychev APPROX '15]. However, due to the connection to Unique Games and hardness of approximation the focus has mostly been on sets of size $k = Ω(n)$, while we focus on the case of general $k$, for which no polylogarithmic approximation is known. We improve the upper bound for this problem by giving an $n^{1/4+\varepsilon}$ approximation for SSBVE for any constant $\varepsilon > 0$. Our algorithm follows in the footsteps of Densest $k$-Subgraph (DkS) and related problems, by designing a tight algorithm for random models, and then extending it to give the same guarantee for arbitrary instances. Moreover, we show that this is tight under plausible complexity conjectures: it cannot be approximated better than $O(n^{1/4})$ assuming an extension of the so-called "Dense vs Random" conjecture for DkS to hypergraphs. We show that the same bound is also matched by an integrality gap for a super-constant number of rounds of the Sherali-Adams LP hierarchy, and an even worse integrality gap for the natural SDP relaxation. Finally, we design a simple bicriteria $\tilde O(\sqrt{n})$ approximation for the more general SSVE problem.

cs.DS↗

arXiv2016

Approximating Spanners and Directed Steiner Forest: Upper and Lower Bounds

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Godwin SODA '16, Godwin-Williams SODA '16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an $O(n^{3/5 + ε})$-approximation for distance preservers and pairwise spanners (for arbitrary constant $ε> 0$). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an $O(\log n)$-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an $O(1)$-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an $O(n^{3/5 + ε})$-approximation for the Directed Steiner Forest problem (for arbitrary constant $ε> 0$) when all edges have uniform costs, improving the previous best $O(n^{2/3 + ε})$-approximation due to Berman et al.~[ICALP '11] (which holds for general edge costs).

cs.DS↗

arXiv2016

The Densest k-Subhypergraph Problem

The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $W\subseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-\sqrt{3})/13 + ε}) \leq O(n^{0.697831+ε})$-approximation (for arbitrary constant $ε> 0$) for Densest $k$-Subhypergraph and an $\tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(\sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.

cs.DS↗

arXiv2016

Computing approximate PSD factorizations

We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.

cs.DS↗

arXiv2015

Smoothed Analysis of Dynamic Networks

We generalize the technique of smoothed analysis to distributed algorithms in dynamic network models. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, dynamic graph smoothed analysis studies the impact of random perturbations of the underlying changing network graph topologies. Similar to the original application of smoothed analysis, our goal is to study whether known strong lower bounds in dynamic network models are robust or fragile: do they withstand small (random) perturbations, or do such deviations push the graphs far enough from a precise pathological instance to enable much better performance? Fragile lower bounds are likely not relevant for real-world deployment, while robust lower bounds represent a true difficulty caused by dynamic behavior. We apply this technique to three standard dynamic network problems with known strong worst-case lower bounds: random walks, flooding, and aggregation. We prove that these bounds provide a spectrum of robustness when subjected to smoothing---some are extremely fragile (random walks), some are moderately fragile / robust (flooding), and some are extremely robust (aggregation).

cs.DC↗

arXiv2015

Explicit Expanding Expanders

Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are "close" to each other. We study the following question: Construct an an infinite sequence of expanders $G_0,G_1,\dots$, such that for every two consecutive graphs $G_i$ and $G_{i+1}$, $G_{i+1}$ can be obtained from $G_i$ by adding a single vertex and inserting/removing a small number of edges, which we call the expansion cost of transitioning from $G_i$ to $G_{i+1}$. This question is very natural, e.g., in the context of datacenter networks, where the vertices represent racks of servers, and the expansion cost captures the amount of rewiring needed when adding another rack to the network. We present an explicit construction of $d$-regular expanders with expansion cost at most $5d/2$, for any $d\geq 6$. Our construction leverages the notion of a "2-lift" of a graph. This operation was first analyzed by Bilu and Linial, who repeatedly applied 2-lifts to construct an infinite family of expanders which double in size from one expander to the next. Our construction can be viewed as a way to "interpolate" between Bilu-Linial expanders with low expansion cost while preserving good edge expansion throughout. While our main motivation is centralized (datacenter networks), we also get the best-known distributed expander construction in the "self-healing" model.

cs.DS↗

arXiv2015

Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+ε)$-resistance sparsifier of size $\tilde O(n/ε)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $Ω(n/ε^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers.

cs.DS↗

arXiv2013

Braess's Paradox in Wireless Networks: The Danger of Improved Technology

When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links). For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decreases the capacity and can in certain cases increase it by $Ω(\log (Δ\cdot P_{\max}))$, where $Δ$ is the ratio of the longest link length to the smallest transmitter-receiver distance and $P_{\max}$ is the maximum transmission power. But there is no reason to expect the optimal capacity to be realized in practice, particularly since maximizing the capacity is known to be NP-hard. In reality, we would expect links to behave as self-interested agents, and thus when introducing a new technology it makes more sense to compare the values reached at game-theoretic equilibria than the optimum values. In this paper we initiate this line of work by comparing various notions of equilibria (particularly Nash equilibria and no-regret behavior) when using a supposedly "better" technology. We show a version of Braess's Paradox for all of them: in certain networks, upgrading technology can actually make the equilibria \emph{worse}, despite an increase in the capacity. We construct instances where this decrease is a constant factor for power control, interference cancellation, and improvements in the SINR threshold ($β$), and is $Ω(\log Δ)$ when power control is combined with interference cancellation. However, we show that these examples are basically tight: the decrease is at most O(1) for power control, interference cancellation, and improved $β$, and is at most $O(\log Δ)$ when power control is combined with interference cancellation.

cs.GT↗

arXiv2012

Matroid Secretary for Regular and Decomposable Matroids

In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g. graphic and transversal matroids). In particular, there is very little known about F-representable matroids (the class of matroids that can be represented as elements of a vector space over a field F), which are one of the foundational matroid classes. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving an O(1)-competitive algorithm for regular matroids (the class of matroids that are representable over every field), and use techniques that are matroid-theoretic rather than graph-theoretic. We use the regular matroid decomposition theorem of Seymour to decompose any regular matroid into matroids which are either graphic, cographic, or isomorphic to R_{10}, and then show how to combine algorithms for these basic classes into an algorithm for regular matroids. This allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids.

cs.DS↗

arXiv2012

Everywhere-Sparse Spanners via Dense Subgraphs

The significant progress in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywhere-sparse (small maximum degree). This disparity is in line with other network design problems, where the maximum-degree objective has been a notorious technical challenge. Our main result is for the Lowest Degree 2-Spanner (LD2S) problem, where the goal is to compute a 2-spanner of an input graph so as to minimize the maximum degree. We design a polynomial-time algorithm achieving approximation factor $\tilde O(Δ^{3-2\sqrt{2}}) \approx \tilde O(Δ^{0.172})$, where $Δ$ is the maximum degree of the input graph. The previous $\tilde O(Δ^{1/4})$ -approximation was proved nearly two decades ago by Kortsarz and Peleg [SODA 1994, SICOMP 1998]. Our main conceptual contribution is to establish a formal connection between LD2S and a variant of the Densest k-Subgraph (DkS) problem. Specifically, we design for both problems strong relaxations based on the Sherali-Adams linear programming (LP) hierarchy, and show that "faithful" randomized rounding of the DkS-variant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle. Unfortunately, the best algorithms known for DkS use the Lovász-Schrijver LP hierarchy in a non-faithful way [Bhaskara, Charikar, Chlamtac, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a subgraph with same log-density.

cs.DS↗

arXiv2012

Label Cover instances with large girth and the hardness of approximating basic k-spanner

We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some $k$, the problem is roughly $2^{\log^{1-ε} n/k}$ hard to approximate for all constant $ε> 0$. A similar theorem was claimed by Elkin and Peleg [ICALP 2000], but their proof was later found to have a fundamental error. We use the new proof to show inapproximability for the basic $k$-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming $NP \not\subseteq BPTIME(2^{polylog(n)})$, we show that for every $k \geq 3$ and every constant $ε> 0$ it is hard to approximate the basic $k$-spanner problem within a factor better than $2^{(\log^{1-ε} n) / k}$ (for large enough $n$). A similar hardness for basic $k$-spanner was claimed by Elkin and Peleg [ICALP 2000], but the error in their analysis of Label Cover made this proof fail as well. Thus for the problem of Label Cover with large girth we give the first non-trivial lower bound. For the basic $k$-spanner problem we improve the previous best lower bound of $Ω(\log n)/k$ by Kortsarz [Algorithmica 1998]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to essentially guarantee large girth.

cs.DS↗