arXiv subjects

Amin Saberi

Publications and source records attributed to Amin Saberi.

At least 55 records · Page 3Linked to original sources

arXiv2022

Improved Online Contention Resolution for Matchings and Applications to the Gig Economy

Motivated by applications in the gig economy, we study approximation algorithms for a \emph{sequential pricing problem}. The input is a bipartite graph $G=(I,J,E)$ between individuals $I$ and jobs $J$. The platform has a value of $v_j$ for matching job $j$ to an individual worker. In order to find a matching, the platform can consider the edges $(i j) \in E$ in any order and make a one-time take-it-or-leave-it offer of a price $π_{ij} = w$ of its choosing to $i$ for completing $j$. The worker accepts the offer with a known probability $ p_{ijw} $; in this case the job and the worker are irrevocably matched. What is the best way to make offers to maximize revenue and/or social welfare? The optimal algorithm is known to be NP-hard to compute (even if there is only a single job). With this in mind, we design efficient approximations to the optimal policy via a new Random-Order Online Contention Resolution Scheme (RO-OCRS) for matching. Our main result is a 0.456-balanced RO-OCRS in bipartite graphs and a 0.45-balanced RO-OCRS in general graphs. These algorithms improve on the recent bound of $\frac{1}{2}(1-e^{-2})\approx 0.432$ of [BGMS21], and improve on the best known lower bounds for the correlation gap of matching, despite applying to a significantly more restrictive setting. As a consequence of our OCRS results, we obtain a $0.456$-approximate algorithm for the sequential pricing problem. We further extend our results to settings where workers can only be contacted a limited number of times, and show how to achieve improved results for this problem, via improved algorithms for the well-studied stochastic probing problem.

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arXiv2021

Beating the Folklore Algorithm for Dynamic Matching

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-Ω(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $ε>0$, no deterministic $(2+ε)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+ε)$-approximate weighted matching algorithm with $O_ε(1)\cdot O(\sqrt[4]{m}) = O_ε(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+ε)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.

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arXiv2021

Algorithms Using Local Graph Features to Predict Epidemics

We study a simple model of epidemics where an infected node transmits the infection to its neighbors independently with probability $p$. This is also known as the independent cascade or Susceptible-Infected-Recovered (SIR) model with fixed recovery time. The size of an outbreak in this model is closely related to that of the giant connected component in ``edge percolation'', where each edge of the graph is kept independently with probability $p$, studied for a large class of networks including configuration model \cite{molloy2011critical} and preferential attachment \cite{bollobas2003,Riordan2005}. Even though these models capture the effects of degree inhomogeneity and the role of super-spreaders in the spread of an epidemic, they only consider graphs that are locally tree like i.e. have a few or no short cycles. Some generalizations of the configuration model were suggested to capture local communities, known as household models \cite{ball2009threshold}, or hierarchical configuration model \cite{Hofstad2015hierarchical}. Here, we ask a different question: what information is needed for general networks to predict the size of an outbreak? Is it possible to make predictions by accessing the distribution of small subgraphs (or motifs)? We answer the question in the affirmative for large-set expanders with local weak limits (also known as Benjamini-Schramm limits). In particular, we show that there is an algorithm which gives a $(1-ε)$ approximation of the probability and the final size of an outbreak by accessing a constant-size neighborhood of a constant number of nodes chosen uniformly at random. We also present corollaries of the theorem for the preferential attachment model, and study generalizations with household (or motif) structure. The latter was only known for the configuration model.

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arXiv2021

Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a "prophet" who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? We study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of $1/2$-competitive algorithms are known against optimum offline. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm which approximates the optimal online algorithm within a factor of $0.51$ -- beating the best-possible prophet inequality. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant $α< 1$.

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arXiv2021

The Stationary Prophet Inequality Problem

We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a $1/2$-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than $(1-1/e)$-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a $1-1/e$ bound implied by recent work of Aouad and Saritaç (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

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arXiv2021

Decentralized Matching in a Probabilistic Environment

We consider a model for repeated stochastic matching where compatibility is probabilistic, is realized the first time agents are matched, and persists in the future. Such a model has applications in the gig economy, kidney exchange, and mentorship matching. We ask whether a $decentralized$ matching process can approximate the optimal online algorithm. In particular, we consider a decentralized $stable$ $matching$ process where agents match with the most compatible partner who does not prefer matching with someone else, and known compatible pairs continue matching in all future rounds. We demonstrate that the above process provides a 0.316-approximation to the optimal online algorithm for matching on general graphs. We also provide a $\frac{1}{7}$-approximation for many-to-one bipartite matching, a $\frac{1}{11}$-approximation for capacitated matching on general graphs, and a $\frac{1}{2k}$-approximation for forming teams of up to $k$ agents. Our results rely on a novel coupling argument that decomposes the successful edges of the optimal online algorithm in terms of their round-by-round comparison with stable matching.

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arXiv2021

The Greedy Algorithm is \emph{not} Optimal for On-Line Edge Coloring

Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $Δ= O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or for random arrival orders, an answer for fully general \emph{adversarial} arrivals remained elusive. We resolve this thirty-year-old conjecture in the affirmative, presenting a $(1.9+o(1))$-competitive online edge coloring algorithm for general graphs of degree $Δ= ω(\log n)$ under vertex arrivals. At the core of our results, and of possible independent interest, is a new online algorithm which rounds a fractional bipartite matching $x$ online under vertex arrivals, guaranteeing that each edge $e$ is matched with probability $(1/2+c)\cdot x_e$, for a constant $c>0.027$.

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arXiv2021

The Value of Excess Supply in Spatial Matching Markets

We study dynamic matching in a spatial setting. Drivers are distributed at random on some interval. Riders arrive in some (possibly adversarial) order at randomly drawn points. The platform observes the location of the drivers, and can match newly arrived riders immediately, or can wait for more riders to arrive. Unmatched riders incur a waiting cost $c$ per period. The platform can match riders and drivers, irrevocably. The cost of matching a driver to a rider is equal to the distance between them. We quantify the value of slightly increasing supply. We prove that when there are $(1+ε)$ drivers per rider (for any $ε> 0$), the cost of matching returned by a simple greedy algorithm which pairs each arriving rider to the closest available driver is $O(\log^3(n))$, where $n$ is the number of riders. On the other hand, with equal number of drivers and riders, even the \emph{ex post} optimal matching does not have a cost less than $Θ(\sqrt{n})$. Our results shed light on the important role of (small) excess supply in spatial matching markets.

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arXiv2021

Tiered Random Matching Markets: Rank is Proportional to Popularity

We study the stable marriage problem in two-sided markets with randomly generated preferences. We consider agents on each side divided into a constant number of "soft tiers", which intuitively indicate the quality of the agent. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the men-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, we show that despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes results of [Pittel 1989] which correspond to uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market, and we give the first explicit calculations of rank beyond uniform markets.

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arXiv2020

Sampling Arborescences in Parallel

We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.

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arXiv2020

Online Hypergraph Matching with Delays

We study an online hypergraph matching problem with delays, motivated by ridesharing applications. In this model, users enter a marketplace sequentially, and are willing to wait up to $d$ timesteps to be matched, after which they will leave the system in favor of an outside option. A platform can match groups of up to $k$ users together, indicating that they will share a ride. Each group of users yields a match value depending on how compatible they are with one another. As an example, in ridesharing, $k$ is the capacity of the service vehicles, and $d$ is the amount of time a user is willing to wait for a driver to be matched to them. We present results for both the utility maximization and cost minimization variants of the problem. In the utility maximization setting, the optimal competitive ratio is $\frac{1}{d}$ whenever $k \geq 3$, and is achievable in polynomial-time for any fixed $k$. In the cost minimization variation, when $k = 2$, the optimal competitive ratio for deterministic algorithms is $\frac{3}{2}$ and is achieved by a polynomial-time thresholding algorithm. When $k>2$, we show that a polynomial-time randomized batching algorithm is $(2 - \frac{1}{d}) \log k$-competitive, and it is NP-hard to achieve a competitive ratio better than $\log k - O (\log \log k)$.

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arXiv2019

Assignment Mechanisms under Distributional Constraints

We study the assignment problem of objects to agents with heterogeneous preferences under distributional constraints. Each agent is associated with a publicly known type and has a private ordinal ranking over objects. We are interested in assigning as many agents as possible. Our first contribution is a generalization of the well-known and widely used serial dictatorship. Our mechanism maintains several desirable properties of serial dictatorship, including strategyproofness, Pareto efficiency, and computational tractability while satisfying the distributional constraints with a small error. We also propose a generalization of the probabilistic serial algorithm, which finds an ordinally efficient and envy-free assignment, and also satisfies the distributional constraints with a small error. We show, however, that no ordinally efficient and envy-free mechanism is also weakly strategyproof. Both of our algorithms assign at least the same number of students as the optimum fractional assignment.

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arXiv2018

Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons

We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in experiments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their $\ell$-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram.

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arXiv2018

Perron-Frobenius Theory in Nearly Linear Time: Positive Eigenvectors, M-matrices, Graph Kernels, and Other Applications

In this paper we provide nearly linear time algorithms for several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative eigenvectors of non-negative matrices, and solving linear systems in asymmetric M-matrices, a generalization of Laplacian systems. The running times of our algorithms depend nearly linearly on the input size and polylogarithmically on the desired accuracy and problem condition number. Leveraging these results we also provide improved running times for a broader range of problems including computing random walk-based graph kernels, computing Katz centrality, and more. The running times of our algorithms improve upon previously known results which either depended polynomially on the condition number of the problem, required quadratic time, or only applied to special cases. We obtain these results by providing new iterative methods for reducing these problems to solving linear systems in Row-Column Diagonally Dominant (RCDD) matrices. Our methods are related to the classic shift-and-invert preconditioning technique for eigenvector computation and constitute the first alternative to the result in Cohen et al. (2016) for reducing stationary distribution computation and solving directed Laplacian systems to solving RCDD systems.

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arXiv2018

Maximum Weight Online Matching with Deadlines

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arXiv2018

Approximation Algorithms for Computing Maximin Share Allocations

We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, so we resort to approximation algorithms. Our main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to $\{0, 1, 2\}$, extending the $\{0, 1\}$ setting studied in Bouveret and Lemaître. We obtain an exact algorithm for any number of agents in this case.

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arXiv2018

Maximizing Efficiency in Dynamic Matching Markets

We study the problem of matching agents who arrive at a marketplace over time and leave after d time periods. Agents can only be matched while they are present in the marketplace. Each pair of agents can yield a different match value, and the planner's goal is to maximize the total value over a finite time horizon. We study matching algorithms that perform well over any sequence of arrivals when there is no a priori information about the match values or arrival times. Our main contribution is a 1/4-competitive algorithm. The algorithm randomly selects a subset of agents who will wait until right before their departure to get matched, and maintains a maximum-weight matching with respect to the other agents. The primal-dual analysis of the algorithm hinges on a careful comparison between the initial dual value associated with an agent when it first arrives, and the final value after d time steps. It is also shown that no algorithm is 1/2-competitive. We extend the model to the case in which departure times are drawn i.i.d from a distribution with non-decreasing hazard rate, and establish a 1/8-competitive algorithm in this setting. Finally we show on real-world data that a modified version of our algorithm performs well in practice.

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arXiv2017

Generating Random Networks Without Short Cycles

Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art in this area, we focus on random graphs without short cycles as a stylized family of graphs, and propose the RandGraph algorithm for randomly generating them. For any constant k, when m=O(n^{1+1/[2k(k+3)]}), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n^2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding $m$ edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research includes introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm, and providing a new method for analyzing such algorithms.

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