arXiv subjects

Jan Vondrak

Publications and source records attributed to Jan Vondrak.

At least 19 recordsLinked to original sources

arXiv2025

Cycle Cancellation for Submodular Fractional Allocations and Applications

We consider discrete allocation problem where $m$ indivisible goods are to be divided among $n$ agents. When agents' valuations are additive, the well-known cycle cancelling lemma by Lenstra, Shmoys, and Tardos plays a key role in design and analysis of rounding algorithms. In this paper, we prove an analogous lemma for the case of submodular valuations. Our algorithm removes cycles in the support graph of a fractional allocation while guaranteeing that each agent's value, measured using the multilinear extension, does not decrease. We demonstrate applications of the cycle-canceling algorithm, along with other ideas, to obtain new algorithms and results for three well-studied allocation objectives: max-min (Santa Claus problem), Nash social welfare (NSW), and maximin-share (MMS). For the submodular NSW problem, we obtain a $\frac{1}{5}$-approximation; for the MMS problem, we obtain a $\frac{1}{2}(1-1/e)$-approximation through new simple algorithms. For various special cases where the goods are "small" valued or the number of agents is constant, we obtain tight/best-known approximation algorithms. All our results are in the value-oracle model.

cs.GT↗

arXiv2024

Prophet Inequalities with Cancellation Costs

Most of the literature on online algorithms in revenue management focuses on settings with irrevocable decisions, where once a decision is made upon the arrival of a new input, it cannot be canceled later. Motivated by modern applications -- such as cloud spot markets, selling banner ads, or online hotel booking -- we introduce and study "prophet inequalities with cancellations" under linear cancellation costs (known as the buyback model). In the classic prophet inequality problem, a sequence of independent random variables $X_1, X_2, \ldots$ with known distributions is revealed one by one, and a decision maker must decide when to stop and accept the current variable in order to maximize the expected value of their choice. In our model, after accepting $X_j$, one may later discard $X_j$ and accept another $X_i$ at a cost of $f \times X_j$, where $f\geq 0$ is a parameter. The goal is to maximize the expected net reward: the value of the final accepted variable minus the total cancellation cost. We aim to design online policies that are competitive against the optimal offline benchmark. Our first main result is an optimal prophet inequality for all parameters $f \ge 0$. We fully characterize the worst-case competitive ratio of the optimal online policy against the optimal offline via the solution to a certain differential equation (for which we provide a constructive solution). Our second main result is to design and analyze a simple and polynomial-time randomized adaptive policy that achieves this optimal competitive ratio. Importantly, our policy is order-agnostic (\`a la [Samuel-Cahn, 1984]), as it only needs the set of distributions and not their arrival order. These results are obtained by novel techniques related to factor-revealing LPs and generalized flow, reductions to a differential equation, and embedding of problem instances into specific Poisson point processes.

cs.DS↗

arXiv2024

Fairness and Incentive Compatibility via Percentage Fees

We study incentive-compatible mechanisms that maximize the Nash Social Welfare. Since traditional incentive-compatible mechanisms cannot maximize the Nash Social Welfare even approximately, we propose changing the traditional model. Inspired by a widely used charging method (e.g., royalties, a lawyer that charges some percentage of possible future compensation), we suggest charging the players some percentage of their value of the outcome. We call this model the percentage fee model. We show that there is a mechanism that maximizes exactly the Nash Social Welfare in every setting with non-negative valuations. Moreover, we prove an analog of Roberts theorem that essentially says that if the valuations are non-negative, then the only implementable social choice functions are those that maximize weighted variants of the Nash Social Welfare. We develop polynomial time incentive compatible approximation algorithms for the Nash Social Welfare with subadditive valuations and prove some hardness results.

cs.GT↗

arXiv2023

A constant factor approximation for Nash social welfare with subadditive valuations

We present a constant-factor approximation algorithm for the Nash social welfare maximization problem with subadditive valuations accessible via demand queries. More generally, we propose a template for NSW optimization by solving a configuration-type LP and using a rounding procedure for (utilitarian) social welfare as a blackbox, which could be applicable to other variants of the problem.

cs.GT↗

arXiv2023

Faster Submodular Maximization for Several Classes of Matroids

The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [ICALP '19]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of $1-1/e-ε$ and both generalize and accelerate the results of Ene and Nguyen [ICALP '19]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [SODA '14]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel FREEZE operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [TALG '05] that maintains the maximum weight basis under insertions and deletions of elements in $O(\log n)$ time. For the transversal matroid the FREEZE operation corresponds to requiring the data structure to keep a certain set $S$ of vertices matched, a property that we call $S$-stability.

cs.DS↗

arXiv2021

Cardinality constrained submodular maximization for random streams

We consider the problem of maximizing submodular functions in single-pass streaming and secretaries-with-shortlists models, both with random arrival order. For cardinality constrained monotone functions, Agrawal, Shadravan, and Stein gave a single-pass $(1-1/e-\varepsilon)$-approximation algorithm using only linear memory, but their exponential dependence on $\varepsilon$ makes it impractical even for $\varepsilon=0.1$. We simplify both the algorithm and the analysis, obtaining an exponential improvement in the $\varepsilon$-dependence (in particular, $O(k/\varepsilon)$ memory). Extending these techniques, we also give a simple $(1/e-\varepsilon)$-approximation for non-monotone functions in $O(k/\varepsilon)$ memory. For the monotone case, we also give a corresponding unconditional hardness barrier of $1-1/e+\varepsilon$ for single-pass algorithms in randomly ordered streams, even assuming unlimited computation. Finally, we show that the algorithms are simple to implement and work well on real world datasets.

cs.DS↗

arXiv2021

Estimating the Nash Social Welfare for coverage and other submodular valuations

We study the Nash Social Welfare problem: Given $n$ agents with valuation functions $v_i:2^{[m]} \rightarrow {\mathbb R}$, partition $[m]$ into $S_1,\ldots,S_n$ so as to maximize $(\prod_{i=1}^{n} v_i(S_i))^{1/n}$. The problem has been shown to admit a constant-factor approximation for additive, budget-additive, and piecewise linear concave separable valuations; the case of submodular valuations is open. We provide a $\frac{1}{e} (1-\frac{1}{e})^2$-approximation of the {\em optimal value} for several classes of submodular valuations: coverage, sums of matroid rank functions, and certain matching-based valuations.

cs.GT↗

arXiv2020

A polynomial lower bound on adaptive complexity of submodular maximization

In large-data applications, it is desirable to design algorithms with a high degree of parallelization. In the context of submodular optimization, adaptive complexity has become a widely-used measure of an algorithm's "sequentiality". Algorithms in the adaptive model proceed in rounds, and can issue polynomially many queries to a function $f$ in each round. The queries in each round must be independent, produced by a computation that depends only on query results obtained in previous rounds. In this work, we examine two fundamental variants of submodular maximization in the adaptive complexity model: cardinality-constrained monotone maximization, and unconstrained non-mono-tone maximization. Our main result is that an $r$-round algorithm for cardinality-constrained monotone maximization cannot achieve an approximation factor better than $1 - 1/e - Ω(\min \{ \frac{1}{r}, \frac{\log^2 n}{r^3} \})$, for any $r < n^c$ (where $c>0$ is some constant). This is the first result showing that the number of rounds must blow up polynomially large as we approach the optimal factor of $1-1/e$. For the unconstrained non-monotone maximization problem, we show a positive result: For every instance, and every $δ>0$, either we obtain a $(1/2-δ)$-approximation in $1$ round, or a $(1/2+Ω(δ^2))$-approximation in $O(1/δ^2)$ rounds. In particular (and in contrast to the cardinality-constrained case), there cannot be an instance where (i) it is impossible to achieve an approximation factor better than $1/2$ regardless of the number of rounds, and (ii) it takes $r$ rounds to achieve a factor of $1/2-O(1/r)$.

cs.DS↗

arXiv2020

Submodular Maximization Through Barrier Functions

In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. More precisely, for maximizing a monotone submodular function subject to the combination of a $k$-matchoid and $\ell$-knapsack constraint (for $\ell\leq k$), we propose a potential function that can be approximately minimized. Once we minimize the potential function up to an $ε$ error it is guaranteed that we have found a feasible set with a $2(k+1+ε)$-approximation factor which can indeed be further improved to $(k+1+ε)$ by an enumeration technique. We extensively evaluate the performance of our proposed algorithm over several real-world applications, including a movie recommendation system, summarization tasks for YouTube videos, Twitter feeds and Yelp business locations, and a set cover problem.

cs.LG↗

arXiv2019

High probability generalization bounds for uniformly stable algorithms with nearly optimal rate

Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousquet and Elisseeff (2002) albeit at the expense of an additional $\sqrt{n}$ factor in the bound. Specifically, their bound on the estimation error of any $γ$-uniformly stable learning algorithm on $n$ samples and range in $[0,1]$ is $O(γ\sqrt{n \log(1/δ)} + \sqrt{\log(1/δ)/n})$ with probability $\geq 1-δ$. The $\sqrt{n}$ overhead makes the bound vacuous in the common settings where $γ\geq 1/\sqrt{n}$. A stronger bound was recently proved by the authors (Feldman and Vondrak, 2018) that reduces the overhead to at most $O(n^{1/4})$. Still, both of these results give optimal generalization bounds only when $γ= O(1/n)$. We prove a nearly tight bound of $O(γ\log(n)\log(n/δ) + \sqrt{\log(1/δ)/n})$ on the estimation error of any $γ$-uniformly stable algorithm. It implies that for algorithms that are uniformly stable with $γ= O(1/\sqrt{n})$, estimation error is essentially the same as the sampling error. Our result leads to the first high-probability generalization bounds for multi-pass stochastic gradient descent and regularized ERM for stochastic convex problems with nearly optimal rate --- resolving open problems in prior work. Our proof technique is new and we introduce several analysis tools that might find additional applications.

cs.LG↗

arXiv2019

Generalization Bounds for Uniformly Stable Algorithms

Uniform stability of a learning algorithm is a classical notion of algorithmic stability introduced to derive high-probability bounds on the generalization error (Bousquet and Elisseeff, 2002). Specifically, for a loss function with range bounded in $[0,1]$, the generalization error of a $γ$-uniformly stable learning algorithm on $n$ samples is known to be within $O((γ+1/n) \sqrt{n \log(1/δ)})$ of the empirical error with probability at least $1-δ$. Unfortunately, this bound does not lead to meaningful generalization bounds in many common settings where $γ\geq 1/\sqrt{n}$. At the same time the bound is known to be tight only when $γ= O(1/n)$. We substantially improve generalization bounds for uniformly stable algorithms without making any additional assumptions. First, we show that the bound in this setting is $O(\sqrt{(γ+ 1/n) \log(1/δ)})$ with probability at least $1-δ$. In addition, we prove a tight bound of $O(γ^2 + 1/n)$ on the second moment of the estimation error. The best previous bound on the second moment is $O(γ+ 1/n)$. Our proofs are based on new analysis techniques and our results imply substantially stronger generalization guarantees for several well-studied algorithms.

cs.LG↗

arXiv2018

Submodular Optimization in the MapReduce Model

Submodular optimization has received significant attention in both practice and theory, as a wide array of problems in machine learning, auction theory, and combinatorial optimization have submodular structure. In practice, these problems often involve large amounts of data, and must be solved in a distributed way. One popular framework for running such distributed algorithms is MapReduce. In this paper, we present two simple algorithms for cardinality constrained submodular optimization in the MapReduce model: the first is a $(1/2-o(1))$-approximation in 2 MapReduce rounds, and the second is a $(1-1/e-ε)$-approximation in $\frac{1+o(1)}ε$ MapReduce rounds.

cs.DC↗

arXiv2018

Concentration of Lipschitz Functions of Negatively Dependent Variables

We explore the question whether Lipschitz functions of random variables under various forms of negative correlation satisfy concentration bounds similar to McDiarmid's inequality for independent random variables. We prove such a concentration bound for random variables satisfying the condition of negative regression, correcting an earlier proof by Dubhashi and Ranjan.

math.PR↗

arXiv2017

Stability and Recovery for Independence Systems

Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial, and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.

cs.DS↗

arXiv2016

Sperner's colorings and optimal partitioning of the simplex

We discuss coloring and partitioning questions related to Sperner's Lemma, originally motivated by an application in hardness of approximation. Informally, we call a partitioning of the $(k-1)$-dimensional simplex into $k$ parts, or a labeling of a lattice inside the simplex by $k$ colors, "Sperner-admissible" if color $i$ avoids the face opposite to vertex $i$. The questions we study are of the following flavor: What is the Sperner-admissible labeling/partitioning that makes the total area of the boundary between different colors/parts as small as possible? First, for a natural arrangement of "cells" in the simplex, we prove an optimal lower bound on the number of cells that must be non-monochromatic in any Sperner-admissible labeling. This lower bound is matched by a simple labeling where each vertex receives the minimum admissible color. Second, we show for this arrangement that in contrast to Sperner's Lemma, there is a Sperner-admissible labeling such that every cell contains at most $4$ colors. Finally, we prove a geometric variant of the first result: For any Sperner-admissible partition of the regular simplex, the total surface area of the boundary shared by at least two different parts is minimized by the Voronoi partition $(A^*_1,\ldots,A^*_k)$ where $A^*_i$ contains all the points whose closest vertex is $i$. We also discuss possible extensions of this result to general polytopes and some open questions.

math.CO↗

arXiv2015

An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles

The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovasz Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovasz Local Lemma and its generalizations, we have designed efficient resampling oracles. As applications of these techniques, we present new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.

cs.DS↗

arXiv2015

Historical Reflections on the Work of IAU Commission 4 (Ephemerides)

As part of a reorganization of the International Astronomical Union (IAU), Commission 4 (Ephemerides) went out of existence after the IAU General Assembly in August 2015. This paper presents brief discussions of some of the developments in fundamental astronomy that have influenced and been influenced by the work of Commission 4 over its 96-year history. The paper also presents notes about some of the publications of the national institutions that have played an essential role in the commission's mission. The contents of this paper were submitted for Commission 4's final report, to appear in IAU Transactions Vol. XXIX-A.

astro-ph.EP↗

arXiv2015

Tight Bounds on Low-degree Spectral Concentration of Submodular and XOS functions

Submodular and fractionally subadditive (or equivalently XOS) functions play a fundamental role in combinatorial optimization, algorithmic game theory and machine learning. Motivated by learnability of these classes of functions from random examples, we consider the question of how well such functions can be approximated by low-degree polynomials in $\ell_2$ norm over the uniform distribution. This question is equivalent to understanding of the concentration of Fourier weight on low-degree coefficients, a central concept in Fourier analysis. We show that 1. For any submodular function $f:\{0,1\}^n \rightarrow [0,1]$, there is a polynomial of degree $O(\log (1/ε) / ε^{4/5})$ approximating $f$ within $ε$ in $\ell_2$, and there is a submodular function that requires degree $Ω(1/ε^{4/5})$. 2. For any XOS function $f:\{0,1\}^n \rightarrow [0,1]$, there is a polynomial of degree $O(1/ε)$ and there exists an XOS function that requires degree $Ω(1/ε)$. This improves on previous approaches that all showed an upper bound of $O(1/ε^2)$ for submodular and XOS functions. The best previous lower bound was $Ω(1/ε^{2/3})$ for monotone submodular functions. Our techniques reveal new structural properties of submodular and XOS functions and the upper bounds lead to nearly optimal PAC learning algorithms for these classes of functions.

cs.DS↗