# Michael Dinitz's research while affiliated with Johns Hopkins University and other places

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## Publications (74)

A $t$-emulator of a graph $G$ is a graph $H$ that approximates its pairwise shortest path distances up to multiplicative $t$ error. We study fault tolerant $t$-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhi...

The best known solutions for $k$-message broadcast in dynamic networks of size $n$ require $\Omega(nk)$ rounds. In this paper, we see if these bounds can be improved by smoothed analysis. We study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every time step, choose a token to broadcast randomly from the...

One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum $k$-Edge-Connected Spanning Subgraph problem ($k$-ECSS), as well as nonuniform demands such as the Survivable Network Design problem. A weakness of these formulations, though, is that we are...

The spread of an epidemic is often modeled by an SIR random process on a social network graph. The MinINF problem for optimal social distancing involves minimizing the expected number of infections, when we are allowed to break at most $B$ edges; similarly the MinINFNode problem involves removing at most $B$ vertices. These are fundamental problems...

A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeof...

A recent line of research investigates how algorithms can be augmented with machine-learned predictions to overcome worst case lower bounds. This area has revealed interesting algorithmic insights into problems, with particular success in the design of competitive online algorithms. However, the question of improving algorithm running times with pr...

There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP. We...

Graph cut problems form a fundamental problem type in combinatorial optimization, and are a central object of study in both theory and practice. In addition, the study of fairness in Algorithmic Design and Machine Learning has recently received significant attention, with many different notions proposed and analyzed in a variety of contexts. In thi...

Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood: the best known lower bound for general $k$ is $\Omega(f^{\...

Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by...

It was recently shown that a version of the greedy algorithm gives a construction of fault-tolerant spanners that is size-optimal, at least for vertex faults. However, the algorithm to construct this spanner is not polynomial-time, and the best-known polynomial time algorithm is significantly suboptimal. Designing a polynomial-time algorithm to con...

New optical technologies offer the ability to reconfigure network topologies dynamically, rather than setting them once and for all. This is true in both optical wide area networks (optical WANs) and in datacenters, despite the many differences between these two settings. Because of these new technologies, there has been a surge of both practical a...

The capacity of wireless networks is a classic and important topic of study. Informally, the capacity of a network is simply the total amount of information which it can transfer. In the context of models of wireless radio networks, this has usually meant the total number of point-to-point messages which can be sent or received in one time step. Th...

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...

We study three capacity problems in the mobile telephone model, a network abstraction that models the peer-to-peer communication capabilities implemented in most commodity smartphone operating systems. The capacity of a network expresses how much sustained throughput can be maintained for a set of communication demands, and is therefore a fundament...

The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree T for graph G, such that the maximum degree of T is the smallest among all spanning trees of G. Let d be this MDST degree for a given graph. In this paper, we present a randomized distributed approximation algorithm for the MDST problem that constructs a sp...

There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP in...

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 Alth\"ofer 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...

The notion of \emph{policy regret} in online learning is a well defined? performance measure for the common scenario of adaptive adversaries, which more traditional quantities such as external regret do not take into account. We revisit the notion of policy regret and first show that there are online learning settings in which policy regret and ext...

Data structures that allow efficient distance estimation (distance oracles or distance sketches) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as the CONGEST model. We initiate their study in newer (and arguably more realistic) models of distributed computation such as t...

In this paper, we generalize the technique of smoothed analysis to apply to distributed algorithms in dynamic networks in which the network graph can change from round to round. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, our proposed dynamic network version o...

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...

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 th...

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 $...

In the Steiner k-Forest problem, we are given an edge weighted graph, a collection D of node pairs, and an integer k ⩽ |D|. The goal is to find a min-weight subgraph that connects at least k pairs. The best known ratio for this problem is min {O(&sqrt;n), O(&sqrt;k)} [Gupta et al. 2010]. In Gupta et al. [2010], it is...

Graph spanners have been studied extensively, and have many applications in algorithms, distributed systems, and computer networks. For many of these application, we want distributed constructions of spanners, i.e., algorithms which use only local information. Dinitz and Krauthgamer (PODC 2011) provided a distributed approximation algorithm for 2-s...

The capacity of wireless networks is a classic and important topic of study. Informally, the capacity of a network is simply the total amount of information which it can transfer. In the context of models of wireless radio networks, this has usually meant the total number of point-to-point messages which can be sent or received in one time step. Th...

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: Const...

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 paramet...

Despite extensive efforts to meet ever-growing demands, today's datacenters often exhibit far-from-optimal performance in terms of network utilization, resiliency to failures, cost efficiency, incremental expandability, and more. Consequently, many novel architectures for high-performance datacenters have been proposed. We show that the benefits of...

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 [Chlamt\'a\v{c} et al APPROX '16]...

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. We show that "sufficiently large" graphs of fixed diameter and degree must be "good" expanders. We...

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 preser...

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 problem...

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(log1-∈ n)/k hard to approximate for all constant ∈ > 0. A similar theorem was claimed by Elkin and Peleg [2000] as part of an attempt to prove hardness for the basic k-sp...

Many architectures for high-performance datacenters have been proposed. Surprisingly, recent studies show that datacenter designs with random network topologies outperform more sophisticated designs, achieving near-optimal throughput and bisection bandwidth, high resiliency to failures, incremental expandability, high cost efficiency, and more. Unf...

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 grap...

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: Const...

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 $(1+\epsilon)$-resistance sparsifier of size $\tilde
O(n/\epsilon)$, and conjecture this bound hol...

A κ-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor κ. We focus on the following problem: Given a graph and a value κ, can we find a κ-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total numbe...

In the Steiner κ-Forest problem we are given an edge weighted graph, a collection D of node pairs, and an integer κ ≤ |D|. The goal is to find a minimum cost subgraph that connects at least κ pairs. The best known ratio for this problem is min{O( √n),O(√κ)} [8]. In [8] it is also shown that ratio ρ for Steiner κ-Forest implies ratio O(ρ · log2 n) f...

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 decrea...

The matroid secretary problem was introduced by Babaioff, Immorlica, and Kleinberg in SODA 2007 as an online problem that was both mathematically interesting and had applications to online auctions. In this column I will introduce and motivate this problem, and give a survey of some of the exciting work that has been done on it over the past 6 year...

In the Packing Interdiction problem we are given a packing LP together with a separate interdiction cost for each LP variable and a global interdiction budget. Our goal is to harm the LP: which variables should we forbid the LP from using (subject to forbidding variables of total interdiction cost at most the budget) in order to minimize the value...

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...

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...

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 $2^{\log^{1-\epsilon} n/k}$ hard to approximate for all
constant $\epsilon > 0$. A similar theorem was claimed by Elkin and Peleg
[ICALP 2000], but their proof was later...

Distance computation is one of the most fundamental primitives used in
communication networks. The cost of effectively and accurately computing
pairwise network distances can become prohibitive in large-scale networks such
as the Internet and Peer-to-Peer (P2P) networks. To negotiate the rising need
for very efficient distance computation, approxim...

We initiate the theoretical study of the problem of minimizing the size of an iBGP (Interior Border Gateway Protocol) overlay in an Autonomous System (AS) in the Internet subject to a natural notion of correctness derived from the standard “hot-potato” routing rules. For both natural versions of the problem (where we measure the size of an overlay...

A natural requirement of many distributed structures is fault-tolerance:
after some failures, whatever remains from the structure should still be
effective for whatever remains from the network. In this paper we examine
spanners of general graphs that are tolerant to vertex failures, and
significantly improve their dependence on the number of fault...

We examine directed spanners through flow-based linear programming relaxations. We design an $\~O(n^{2/3})$-approximation algorithm for the directed $k$-spanner problem that works for all $k\geq 1$, which is the first sublinear approximation for arbitrary edge-lengths. Even in the more restricted setting of unit edge-lengths, our algorithm improves...

In this paper we consider the problem of maximizing wireless network capacity (a.k.a. one-shot scheduling) in both the protocol and physical models. We give the first distributed algorithms with provable guarantees in the physical model, and show how they can be generalized to more complicated metrics and settings in which the physical assumptions...

In this work we study a variety of problems, including network coordinate systems, compact routing, and wireless network capacity. The unifying thread is the observation that while strong theoretical properties are already known about all of these problems, the standard models used to prove these properties are not particularly realistic. We attemp...

In this paper we consider the problem of maximizing the number of supported connections in arbitrary wireless networks where a transmission is supported if and only if the signal-to-interference-plus-noise ratio at the receiver is greater than some threshold. The aim is to choose transmission powers for each connection so as to maximize the number...

The classical secretary problem studies the problem of selecting online an element (a "secretary") with maximum value in a randomly ordered sequence. The difficulty lies in the fact that an element must be either selected or discarded upon its arrival, and this decision is irrevocable. Constant-competitive algorithms are known for the classical sec...

In this paper we consider the problem of maximizing wireless network capacity (a.k.a. one-shot scheduling) in both the protocol and physical models. We give the first distributed algorithms with provable guarantees in the physical model, and also give the first algorithms in the protocol model that do not assume transmitters can coordinate with the...

The theoretical computer science community has traditionally used embeddings of finite metrics as a tool in designing approximation algorithms. Recently, however, there has been considerable interest in using metric embeddings in the context of networks to allow network nodes to have more knowledge of the pairwise distances between other nodes in t...

In many network applications it would be useful for individual nodes to have knowledge of distances (e.g. latency or bandwidth) in the network. If every node stored all distances then it would take Ω(n2) space at every node, and furthermore when a new node joins it would have to send an Ω(n)-bit message to Ω(n) other nodes, for a total communicatio...

Given a weighted graph G=(V,E), a compact routing scheme is a distributed algorithm for forwarding packets from any source to any destination. The fundamental tradeoff is between the space used at each node and the stretch of the total route, measured by the multiplicative factor between the actual distance traveled and the length of the shortest p...

Given a metric (V,d), a spanner is a sparse graph whose shortest-path metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distance...

A tiling of a finite abelian group $G$ is a pair $(V,A)$ of subsets of $G$ such that $0$ is in both $V$ and $A$ and every $g\in G$ can be uniquely written as $g=v+a$ with $v\in V$ and $a\in A$. Tilings are a special case of normed factorizations, a type of factorization by subsets that was introduced by Hajós [Casopsis Puest Path. Rys., 74, (1949),...

## Citations

... We note even the (fault-free) Baswana-Sen algorithm provides (2k − 1) spanner with O(kn 1+1/k ) edges. Providing optimal algorithms for optimal EFT-spanners is also a major open problem, in light of the recent work by Bodwin, Dinitz and Robelle [BDR21b]. ...

... Dinitz, Nazari and Zhang [79] prove a stronger integrality gap than Theorem 9.2 by considering Lasserre lifts of the flow LP. They essentially show that even the strongest lift and project methods cannot help significantly in the approximation of spanners. ...

Reference: Graph spanners: A tutorial review

... We construct a subgraph with O(2 f k+f +k k · n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F . This improves significantly over the O(f n There is a large amount of research in computer science that is devoted to the problems of computing shortest paths and distances in graphs that are subject to a small number of transient failures (see [1,6,7,8,18,19,26,27,28,30,34] and the references therein). These problems usually ask to compute suitable subgraphs, a.k.a. ...

Reference: Fixed-Parameter Sensitivity Oracles

... Recon-figurable demand-aware networks may also rely on expander graphs, e.g., Flexspander [67] and Kevin [73], and are currently also considered as a promising solution to speed up data transfers in supercomputers [35,24]. The notion of demand-aware networks raise novel optimization problems related to switch scheduling [49], and recently interesting first insights have been obtained both for offline [68] and for online scheduling [62,23,43,10]. Due to the increased reconfiguration time experienced in demand-aware networks, many existing demand-aware architectures additionally rely on a fixed network. ...

... There has been a fruitful line of work tackling this question for spanners and emulators (e.g., [6][7][8][9][10][11][12][13]16]). This has produced optimal bounds on the price of fault tolerance in some settings, distinctions between edge and vertex faults, and a suite of algorithmic techniques that allow for efficient computation of sparse fault tolerant spanners/emulators (including in distributed and parallel models of computation). ...

... Roughly speaking, we can use our spectral results to show that if a regular graph or hypergraph has a small second eigenvalue τ 2 (this is described precisely later in the paper), then it cannot have the largest order. This result is an extension of a result in [22]. Moore polygons [20] are optimal graphs in regard to the problems (1) and (2) for r-regular simple graphs as proved in [14,39]. ...

... The last session of the conference included several talks on distributed network design algorithms. The speakers presented efficient distributed algorithms that build sparse backbones satisfying good properties, as preserving distances [28,13], having small maximum degree [24] or resistance to failures [13,25]. Michael Dinitz gave a talk on distributed minimum degree spanning trees [24]. ...

Reference: PODC 2019 Review

... A number of other network design problems exhibit behavior that is similar to spanners, and we can extend our integrality gaps to these problems. In particular, we give a new integrality gap for Lasserre for Directed Steiner Network (DSN) (also called Directed Steiner Forest) and Shallow-Light Steiner Network (SLSN) [2]. In DSN we are given a directed graph G = (V, E) (possibly with weights) and a collection of pairs {(s i , t i )} i∈ [p] , and are asked to find the cheapest subgraph such that there is a s i to t i path for all i ∈ [p]. ...

... In particular, they showed that the problem can be solved with both subcubic preprocessing time and subquadratic query time. 8 An obvious open problem is whether one can achieve subcubic preprocessing time with linear (or even sublinear) query time. We answer this question affirmatively. ...

... It might be a priori surprising that sparse FT spanners exist, but Chechik et al. [22] gave the first construction of d-FT (2k − 1)-spanners with O(d 2 k d+1 · n 1+1/k log 1−1/k n) edges. Subsequent papers [17,19,34] improved the number of edges to O(n 1+1/k d 1−1/k ), which is optimal assuming the girth conjecture of Erdős [39]. Besides FT spanners, there are many other kinds of fault-tolerant structures, e.g. ...