arXiv subjects

Michael Dinitz

Publications and source records attributed to Michael Dinitz.

At least 37 records · Page 2Linked to original sources

arXiv2023

Degrees and Network Design: New Problems and Approximations

While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints which might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the $\ell_p$-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the $\ell_{\infty}$-norm) and the number of edges (the $\ell_1$-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogrithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Kortsarz and Nutov, Discret. Appl. Math. 2022] and [Guo et al., Algorithmica 2022].

cs.DS↗

arXiv2022

Algorithms with Prediction Portfolios

The research area of algorithms with predictions has seen recent success showing how to incorporate machine learning into algorithm design to improve performance when the predictions are correct, while retaining worst-case guarantees when they are not. Most previous work has assumed that the algorithm has access to a single predictor. However, in practice, there are many machine learning methods available, often with incomparable generalization guarantees, making it hard to pick a best method a priori. In this work we consider scenarios where multiple predictors are available to the algorithm and the question is how to best utilize them. Ideally, we would like the algorithm's performance to depend on the quality of the best predictor. However, utilizing more predictions comes with a cost, since we now have to identify which prediction is the best. We study the use of multiple predictors for a number of fundamental problems, including matching, load balancing, and non-clairvoyant scheduling, which have been well-studied in the single predictor setting. For each of these problems we introduce new algorithms that take advantage of multiple predictors, and prove bounds on the resulting performance.

cs.LG↗

arXiv2022

Epic Fail: Emulators can tolerate polynomially many edge faults for free

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 exhibit surprisingly different behavior. In particular, our main result is that, for $(2k-1)$-emulators with $k$ odd, we can tolerate a polynomial number of edge faults for free. For example: for any $n$-node input graph, we construct a $5$-emulator ($k=3$) on $O(n^{4/3})$ edges that is robust to $f = O(n^{2/9})$ edge faults. It is well known that $Ω(n^{4/3})$ edges are necessary even if the $5$-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd $k$, and whether a similar phenomenon can be proved for even $k$.

cs.DS↗

arXiv2022

Smoothed Analysis of Information Spreading in Dynamic Networks

The best known solutions for $k$-message broadcast in dynamic networks of size $n$ require $Ω(nk)$ 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 set of tokens you know. We show that with even a small amount of smoothing (one random edge added per round), this natural strategy solves $k$-message broadcast in $\tilde{O}(n+k^3)$ rounds, with high probability, beating the best known bounds for $k=o(\sqrt{n})$ and matching the $Ω(n+k)$ lower bound for static networks for $k=O(n^{1/3})$ (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: given $\ell$-smoothing (i.e., $\ell$ random edges added per round), this simple strategy terminates in $O(kn^{2/3}\log^{1/3}(n)\ell^{-1/3})$ rounds. We then prove this analysis close to tight with an almost-matching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of $\tilde{O}(k\sqrt{n})$ rounds to solve $k$-message broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply our tools to prove an optimal result for $k$-message broadcast in so-called well-mixed networks in the absence of smoothing. By comparing this result to an existing lower bound for well-mixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to well-mixed token spreading, partially resolving an open question on the impact of adversary strength on the $k$-message broadcast problem.

cs.DC↗

arXiv2022

Relative Survivable Network Design

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 not able to ask for fault-tolerance larger than the connectivity. We introduce and study new variants of these problems under a notion of relative fault-tolerance. Informally, we require not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. That is, the subgraph we build must "behave" identically to the underlying graph with respect to connectivity after bounded faults. We define and introduce these problems, and provide the first approximation algorithms: a $(1+4/k)$-approximation for the unweighted relative version of $k$-ECSS, a $2$-approximation for the weighted relative version of $k$-ECSS, and a $27/4$-approximation for the special case of Relative Survivable Network Design with only a single demand with a connectivity requirement of $3$. To obtain these results, we introduce a number of technical ideas that may of independent interest. First, we give a generalization of Jain's iterative rounding analysis that works even when the cut-requirement function is not weakly supermodular, but instead satisfies a weaker definition we introduce and term local weak supermodularity. Second, we prove a structure theorem and design an approximation algorithm utilizing a new decomposition based on important separators, which are structures commonly used in fixed-parameter algorithms that have not commonly been used in approximation algorithms.

cs.DS↗

arXiv2022

Fair Disaster Containment via Graph-Cut Problems

Graph cut problems are fundamental in Combinatorial Optimization, and are a central object of study in both theory and practice. Furthermore, the study of \emph{fairness} in Algorithmic Design and Machine Learning has recently received significant attention, with many different notions proposed and analyzed for a variety of contexts. In this paper we initiate the study of fairness for graph cut problems by giving the first fair definitions for them, and subsequently we demonstrate appropriate algorithmic techniques that yield a rigorous theoretical analysis. Specifically, we incorporate two different notions of fairness, namely \emph{demographic} and \emph{probabilistic individual} fairness, in a particular cut problem that models disaster containment scenarios. Our results include a variety of approximation algorithms with provable theoretical guarantees.

cs.DS↗

arXiv2022

Controlling Epidemic Spread using Probabilistic Diffusion Models on Networks

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 in epidemiology and network science. While a number of heuristics have been considered, the complexity of these problems remains generally open. In this paper, we present two bicriteria approximation algorithms for MinINF, which give the first non-trivial approximations for this problem. The first is based on the cut sparsification result of Karger \cite{karger:mathor99}, and works when the transmission probabilities are not too small. The second is a Sample Average Approximation (SAA) based algorithm, which we analyze for the Chung-Lu random graph model. We also extend some of our results to tackle the MinINFNode problem.

cs.DS↗

arXiv2021

Vertex Fault-Tolerant Emulators

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 tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $\log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

cs.DS↗

arXiv2021

Faster Matchings via Learned Duals

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 predictions has largely been unexplored. We take a first step in this direction by combining the idea of machine-learned predictions with the idea of "warm-starting" primal-dual algorithms. We consider one of the most important primitives in combinatorial optimization: weighted bipartite matching and its generalization to $b$-matching. We identify three key challenges when using learned dual variables in a primal-dual algorithm. First, predicted duals may be infeasible, so we give an algorithm that efficiently maps predicted infeasible duals to nearby feasible solutions. Second, once the duals are feasible, they may not be optimal, so we show that they can be used to quickly find an optimal solution. Finally, such predictions are useful only if they can be learned, so we show that the problem of learning duals for matching has low sample complexity. We validate our theoretical findings through experiments on both real and synthetic data. As a result we give a rigorous, practical, and empirically effective method to compute bipartite matchings.

cs.LG↗

arXiv2021

Partially Optimal Edge Fault-Tolerant Spanners

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 $Ω(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} +fn)$. Our main result is to nearly close this gap with an improved upper bound, thus separating the cases of edge and vertex faults. For odd $k$, our new upper bound is $O_k(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} + fn)$, which is tight up to hidden $poly(k)$ factors. For even $k$, our new upper bound is $O_k(f^{1/2} n^{1+1/k} +fn)$, which leaves a gap of $poly(k) f^{1/(2k)}$. Our proof is an analysis of the fault-tolerant greedy algorithm, which requires exponential time, but we also show that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of $k$.

cs.DS↗

arXiv2020

Optimal Vertex Fault-Tolerant Spanners in Polynomial Time

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 analyzing the output spanner of a certain exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes $\widetilde{O}\left( f^{1-1/k} n^{2+1/k} + mf^2\right)$ time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [Bodwin-Patel PODC '19], the only previously known algorithm for constructing optimal vertex fault-tolerant spanners.

cs.DS↗

arXiv2020

Efficient and Simple Algorithms for Fault Tolerant Spanners

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 construct (near-)optimal fault-tolerant spanners was given as an explicit open problem in the two most recent papers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka Williams SODA '18] and [Bodwin, Patel PODC '19]). We give a surprisingly simple algorithm which runs in polynomial time and constructs fault-tolerant spanners that are extremely close to optimal (off by only a linear factor in the stretch) by modifying the greedy algorithm to run in polynomial time. To complement this result, we also give simple distributed constructions in both the LOCAL and CONGEST models.

cs.DS↗

arXiv2020

Policy Regret in Repeated Games

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 external regret are incompatible: any sequence of play that achieves a favorable regret with respect to one definition must do poorly with respect to the other. We then focus on the game-theoretic setting where the adversary is a self-interested agent. In that setting, we show that external regret and policy regret are not in conflict and, in fact, that a wide class of algorithms can ensure a favorable regret with respect to both definitions, so long as the adversary is also using such an algorithm. We also show that the sequence of play of no-policy regret algorithms converges to a \emph{policy equilibrium}, a new notion of equilibrium that we introduce. Relating this back to external regret, we show that coarse correlated equilibria, which no-external regret players converge to, are a strict subset of policy equilibria. Thus, in game-theoretic settings, every sequence of play with no external regret also admits no policy regret, but the converse does not hold.

cs.LG↗

arXiv2020

Scheduling for Weighted Flow and Completion Times in Reconfigurable Networks

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 and theoretical research on algorithms to take advantage of them. In particular, Jia et al. [INFOCOM '17] designed online scheduling algorithms for dynamically reconfigurable topologies for both the makespan and sum of completion times objectives. In this paper, we work in the same setting but study an objective that is more meaningful in an online setting: the sum of flow times. The flow time of a job is the total amount of time that it spends in the system, which may be considerably smaller than its completion time if it is released late. We provide competitive algorithms for the online setting with speed augmentation, and also give a lower bound proving that speed augmentation is in fact necessary. As a side effect of our techniques, we also improve and generalize the results of Jia et al. on completion times by giving an $O(1)$-competitive algorithm for arbitrary sizes and release times even when nodes have different degree bounds, and moreover allow for the weighted sum of completion times (or flow times).

cs.DS↗

arXiv2019

Reception Capacity: Definitions, Game Theory and Hardness

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. This definition has seen intensive study in recent years, particularly with respect to more accurate models of radio networks such as the SINR model. This paper is motivated by an obvious fact: radio antennae are (at least traditionally) omnidirectional, and hence point-to-point connections are not necessarily the best definition of the true capacity of a wireless network. To fix this, we introduce a new definition of reception capacity as the maximum number of messages which can be received in one round, and show that this is related to a new optimization problem we call the Maximum Perfect Dominated Set (MaxPDS) problem. Using this relationship we give a tight lower bound for approximating this capacity which essentially matches a known upper bound. As our main result, we analyze this notion of capacity under game-theoretic constraints, giving tight bounds on the average quality achieved at any coarse correlated equilibrium (and thus at any Nash). This immediately gives bounds on the average behavior of the natural distributed algorithm in which every transmitter uses online learning algorithms to learn whether to transmit.

cs.GT↗

arXiv2019

Massively Parallel Approximate Distance Sketches

Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. We provide efficient constructions in both of these models, but our core results are for MPC. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds. Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use to construct distance sketches are an MPC construction of the hopsets of Elkin and Neiman (2016). This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting.

cs.DS↗

arXiv2019

The Capacity of Smartphone Peer-to-Peer Networks

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 fundamental bound on the usefulness of a network. Because of this importance, wireless network capacity has been active area of research for the last two decades. The three capacity problems that we study differ in the structure of the communication demands. The first problem is pairwise capacity, where the demands are (source, destination) pairs. Pairwise capacity is one of the most classical definitions, as it was analyzed in the seminal paper of Gupta and Kumar on wireless network capacity. The second problem we study is broadcast capacity, in which a single source must deliver packets to all other nodes in the network. Finally, we turn our attention to all-to-all capacity, in which all nodes must deliver packets to all other nodes. In all three of these problems we characterize the optimal achievable throughput for any given network, and design algorithms which asymptotically match this performance. We also study these problems in networks generated randomly by a process introduced by Gupta and Kumar, and fully characterize their achievable throughput. Interestingly, the techniques that we develop for all-to-all capacity also allow us to design a one-shot gossip algorithm that runs within a polylogarithmic factor of optimal in every graph. This largely resolves an open question from previous work on the one-shot gossip problem in this model.

cs.DC↗

arXiv2019

Lasserre Integrality Gaps for Graph Spanners and Related Problems

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 variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for $n^{Ω(ε)}$ levels of the Lasserre hierarchy, for both the directed and undirected spanner problems. We also extend these integrality gaps to related problems, notably Directed Steiner Network and Shallow-Light Steiner Network.

cs.CC↗