Searcharxiv⌕ Search

arXiv subjects

Stefano Leonardi

Publications and source records attributed to Stefano Leonardi.

54 records · Page 3Linked to original sources
arXiv2021

Budget Feasible Mechanisms on Matroids

Motivated by many practical applications, in this paper we study {\em budget feasible mechanisms} where the goal is to procure independent sets from matroids. More specifically, we are given a matroid $\mathcal{M}=(E,\mathcal{I})$ where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements $E$. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with $4$-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns $α-$approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with $(3α+1)-$approximation to the optimal common independent set.

cs.GT↗
arXiv2021

Fair Clustering with Multiple Colors

A fair clustering instance is given a data set $A$ in which every point is assigned some color. Colors correspond to various protected attributes such as sex, ethnicity, or age. A fair clustering is an instance where membership of points in a cluster is uncorrelated with the coloring of the points. Of particular interest is the case where all colors are equally represented. If we have exactly two colors, Chierrichetti, Kumar, Lattanzi and Vassilvitskii (NIPS 2017) showed that various $k$-clustering objectives admit a constant factor approximation. Since then, a number of follow up work has attempted to extend this result to a multi-color case, though so far, the only known results either result in no-constant factor approximation, apply only to special clustering objectives such as $k$-center, yield bicrititeria approximations, or require $k$ to be constant. In this paper, we present a simple reduction from unconstrained $k$-clustering to fair $k$-clustering for a large range of clustering objectives including $k$-median, $k$-means, and $k$-center. The reduction loses only a constant factor in the approximation guarantee, marking the first true constant factor approximation for many of these problems.

cs.DS↗
arXiv2021

A Regret Analysis of Bilateral Trade

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. Despite the simplicity of this problem, a classical result by Myerson and Satterthwaite (1983) affirms the impossibility of designing a mechanism which is simultaneously efficient, incentive compatible, individually rational, and budget balanced. This impossibility result fostered an intense investigation of meaningful trade-offs between these desired properties. Much work has focused on approximately efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016), Colini-Baldeschi et al. (2016), which have been shown to fully characterize strong budget balanced and ex-post individually rational direct revelation mechanisms. All these results, however, either assume some knowledge on the priors of the seller/buyer valuations, or a black box access to some samples of the distributions, as in D{ü}tting et al. (2021). In this paper, we cast for the first time the bilateral trade problem in a regret minimization framework over rounds of seller/buyer interactions, with no prior knowledge on the private seller/buyer valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different models of feedback and private valuations, using as benchmark the best fixed price in hindsight. More precisely, we prove the following bounds on the regret: $\bullet$ $\widetildeΘ(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms); $\bullet$ $\widetildeΘ(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and seller/buyer valuations with bounded densities; $\bullet$ $Θ(T)$ for realistic feedback and independent seller/buyer valuations; $\bullet$ $Θ(T)$ for the adversarial setting.

cs.LG↗
arXiv2020

Pandora's Box Problem with Order Constraints

The Pandora's Box Problem, originally formalized by Weitzman in 1979, models selection from set of random, alternative options, when evaluation is costly. This includes, for example, the problem of hiring a skilled worker, where only one hire can be made, but the evaluation of each candidate is an expensive procedure. Weitzman showed that the Pandora's Box Problem admits an elegant, simple solution, where the options are considered in decreasing order of reservation value,i.e., the value that reduces to zero the expected marginal gain for opening the box. We study for the first time this problem when order - or precedence - constraints are imposed between the boxes. We show that, despite the difficulty of defining reservation values for the boxes which take into account both in-depth and in-breath exploration of the various options, greedy optimal strategies exist and can be efficiently computed for tree-like order constraints. We also prove that finding approximately optimal adaptive search strategies is NP-hard when certain matroid constraints are used to further restrict the set of boxes which may be opened, or when the order constraints are given as reachability constraints on a DAG. We complement the above result by giving approximate adaptive search strategies based on a connection between optimal adaptive strategies and non-adaptive strategies with bounded adaptivity gap for a carefully relaxed version of the problem.

cs.GT↗
arXiv2020

Algorithms for Hiring and Outsourcing in the Online Labor Market

Although freelancing work has grown substantially in recent years, in part facilitated by a number of online labor marketplaces, (e.g., Guru, Freelancer, Amazon Mechanical Turk), traditional forms of "in-sourcing" work continue being the dominant form of employment. This means that, at least for the time being, freelancing and salaried employment will continue to co-exist. In this paper, we provide algorithms for outsourcing and hiring workers in a general setting, where workers form a team and contribute different skills to perform a task. We call this model team formation with outsourcing. In our model, tasks arrive in an online fashion: neither the number nor the composition of the tasks is known a-priori. At any point in time, there is a team of hired workers who receive a fixed salary independently of the work they perform. This team is dynamic: new members can be hired and existing members can be fired, at some cost. Additionally, some parts of the arriving tasks can be outsourced and thus completed by non-team members, at a premium. Our contribution is an efficient online cost-minimizing algorithm for hiring and firing team members and outsourcing tasks. We present theoretical bounds obtained using a primal-dual scheme proving that our algorithms have a logarithmic competitive approximation ratio. We complement these results with experiments using semi-synthetic datasets based on actual task requirements and worker skills from three large online labor marketplaces.

math.OC↗
arXiv2020

Algorithms for Fair Team Formation in Online Labour Marketplaces

As freelancing work keeps on growing almost everywhere due to a sharp decrease in communication costs and to the widespread of Internet-based labour marketplaces (e.g., guru.com, feelancer.com, mturk.com, upwork.com), many researchers and practitioners have started exploring the benefits of outsourcing and crowdsourcing. Since employers often use these platforms to find a group of workers to complete a specific task, researchers have focused their efforts on the study of team formation and matching algorithms and on the design of effective incentive schemes. Nevertheless, just recently, several concerns have been raised on possibly unfair biases introduced through the algorithms used to carry out these selection and matching procedures. For this reason, researchers have started studying the fairness of algorithms related to these online marketplaces, looking for intelligent ways to overcome the algorithmic bias that frequently arises. Broadly speaking, the aim is to guarantee that, for example, the process of hiring workers through the use of machine learning and algorithmic data analysis tools does not discriminate, even unintentionally, on grounds of nationality or gender. In this short paper, we define the Fair Team Formation problem in the following way: given an online labour marketplace where each worker possesses one or more skills, and where all workers are divided into two or more not overlapping classes (for examples, men and women), we want to design an algorithm that is able to find a team with all the skills needed to complete a given task, and that has the same number of people from all classes. We provide inapproximability results for the Fair Team Formation problem together with four algorithms for the problem itself. We also tested the effectiveness of our algorithmic solutions by performing experiments using real data from an online labor marketplace.

cs.CY↗
arXiv2019

Online Revenue Maximization for Server Pricing

Efficient and truthful mechanisms to price resources on remote servers/machines has been the subject of much work in recent years due to the importance of the cloud market. This paper considers revenue maximization in the online stochastic setting with non-preemptive jobs and a unit capacity server. One agent/job arrives at every time step, with parameters drawn from an underlying unknown distribution. We design a posted-price mechanism which can be efficiently computed, and is revenue-optimal in expectation and in retrospect, up to additive error. The prices are posted prior to learning the agent's type, and the computed pricing scheme is deterministic, depending only on the length of the allotted time interval and on the earliest time the server is available. If the distribution of agent's type is only learned from observing the jobs that are executed, we prove that a polynomial number of samples is sufficient to obtain a near-optimal truthful pricing strategy.

cs.GT↗
arXiv2019

Envy, Regret, and Social Welfare Loss

Incentive compatibility (IC) is one of the most fundamental properties of an auction mechanism, including those used for online advertising. Recent methods by Feng et al. and Lahaie et al. show that counterfactual runs of the auction mechanism with different bids can be used to determine whether an auction is IC. In this paper we show that a similar result can be obtained by looking at the advertisers' envy, which can be computed with one single execution of the auction. We introduce two metrics to evaluate the incentive-compatibility of an auction: IC-Regret and IC-Envy. For position auction environments, we show that for a large class of pricing schemes (which includes e.g. VCG and GSP), IC-Envy $\ge$ IC-Regret (and IC-Envy = IC-Regret when bids are distinct). We consider non-separable discounts in the Ad Types environment of Colini-Baldeschi et al. where we show that for a generalization of GSP also IC-Envy $\ge$ IC-Regret. Our final theoretical result is that in all these settings IC-Envy be used to bound the loss in social welfare due advertiser misreports. Finally, we show that IC-Envy is useful as a feature to predict IC-Regret in auction environments beyond the ones for which we show theoretical results. In particular, using IC-Envy yields better results than training models using only price and value features.

cs.GT↗
arXiv2017

Fixed Price Approximability of the Optimal Gain From Trade

Bilateral trade is a fundamental economic scenario comprising a strategically acting buyer and seller, each holding valuations for the item, drawn from publicly known distributions. A mechanism is supposed to facilitate trade between these agents, if such trade is beneficial. It was recently shown that the only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, are fixed price mechanisms, i.e., mechanisms that are parametrised by a price p, and trade occurs if and only if the valuation of the buyer is at least p and the valuation of the seller is at most p. The gain from trade is the increase in welfare that results from applying a mechanism; here we study the gain from trade achievable by fixed price mechanisms. We explore this question for both the bilateral trade setting, and a double auction setting where there are multiple buyers and sellers. We first identify a fixed price mechanism that achieves a gain from trade of at least 2/r times the optimum, where r is the probability that the seller's valuation does not exceed the buyer's valuation. This extends a previous result by McAfee. Subsequently, we improve this approximation factor in an asymptotic sense, by showing that a more sophisticated rule for setting the fixed price results in an expected gain from trade within a factor O(log(1/r)) of the optimal gain from trade. This is asymptotically the best approximation factor possible. Lastly, we extend our study of fixed price mechanisms to the double auction setting defined by a set of multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. We present a fixed price mechanism that achieves a gain from trade that achieves for all epsilon > 0 a gain from trade of at least (1-epsilon) times the expected optimal gain from trade with probability 1 - 2/e^{#T epsilon^2 /2}, where #T is the expected number of trades resulting from the double auction.

cs.GT↗
arXiv2017

When the Optimum is also Blind: a New Perspective on Universal Optimization

Consider the following variant of the set cover problem. We are given a universe $U=\{1,...,n\}$ and a collection of subsets $\mathcal{C} = \{S_1,...,S_m\}$ where $S_i \subseteq U$. For every element $u \in U$ we need to find a set $ϕ(u) \in \mathcal C$ such that $u\in ϕ(u)$. Once we construct and fix the mapping $ϕ:U \rightarrow \mathcal{C}$ a subset $X \subseteq U$ of the universe is revealed, and we need to cover all elements from $X$ with exactly $ϕ(X):=\cup_{u\in X} ϕ(u)$. The goal is to find a mapping such that the cover $ϕ(X)$ is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate the approximability of such problems under the regime of worst case analysis or when the input instance is drawn randomly from some probability distribution. Here one typically compares the quality of the produced solution with the optimal offline solution. In this paper we consider a different viewpoint: What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint it is possible to achieve improved approximation algorithms for the stochastic version of universal set cover. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms also for other related problems.

cs.CC↗
arXiv2016

Revenue Maximizing Envy-Free Pricing in Matching Markets with Budgets

We study envy-free pricing mechanisms in matching markets with $m$ items and $n$ budget constrained buyers. Each buyer is interested in a subset of the items on sale, and she appraises at some single-value every item in her preference-set. Moreover, each buyer has a budget that constraints the maximum affordable payment, while she aims to obtain as many items as possible of her preference-set. Our goal is to compute an envy-free pricing allocation that maximizes the revenue, i.e., the total payment charged to the buyers. This pricing problem is hard to approximate better than $Ω({\rm min} \{n,m\}^{1/2-ε})$ for any $ε>0$, unless $P=NP$. The hardness result is due to the presence of the matching constraints given that the simpler multi-unit case can be approximated up to a constant factor of $2$. The goal of this paper is to circumvent the hardness result by restricting ourselves to specific settings of valuations and budgets. Two particularly significant scenarios are: each buyer has a budget that is greater than her single-value valuation, and each buyer has a budget that is lower than her single-value valuation. Surprisingly, in both scenarios we are able to achieve a $1/4$-approximation to the optimal envy-free revenue. The algorithms utilize a novel version of the Ausebel ascending price auction. These results may suggest that, although it is difficult to approximate the optimal revenue in general, ascending price auctions could achieve relatively good revenue in most of the practical settings.

cs.GT↗
arXiv2015

Sequential Posted Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a take-it-or-leave-it offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the the case of a continuous valuation distribution when various standard assumptions hold simultaneously. In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to ratio of the highest and lowest possible valuation. We prove that for two simple generalizations of these mechanisms, a better revenue performance can be achieved: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Omega(1/max{1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d=0) to arbitrary dependence (d=n-1). Moreover, when we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer that depends on the valuations of all buyers except i's, we prove that a constant fraction (2-sqrt{e})/4~0.088 of the optimal revenue can be always be extracted.

cs.GT↗
arXiv2012

Near-Optimal Multi-Unit Auctions with Ordered Bidders

We construct prior-free auctions with constant-factor approximation guarantees with ordered bidders, in both unlimited and limited supply settings. We compare the expected revenue of our auctions on a bid vector to the monotone price benchmark, the maximum revenue that can be obtained from a bid vector using supply-respecting prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. As a consequence, our auctions are simultaneously near-optimal in a wide range of Bayesian multi-unit environments.

cs.GT↗
arXiv2012

A Mazing 2+eps Approximation for Unsplittable Flow on a Path

We study the unsplittable flow on a path problem (UFP) where we are given a path with non-negative edge capacities and tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. This problem naturally arises in many settings such as bandwidth allocation, resource constrained scheduling, and interval packing. A natural task classification defines the size of a task i to be the ratio delta between the demand of i and the minimum capacity of any edge used by i. If all tasks have sufficiently small delta, the problem is already well understood and there is a 1+eps approximation. For the complementary setting---instances whose tasks all have large delta---much remains unknown, and the best known polynomial-time procedure gives only (for any constant delta>0) an approximation ratio of 6+eps. In this paper we present a polynomial time 1+eps approximation for the latter setting. Key to this result is a complex geometrically inspired dynamic program. Here each task is represented as a segment underneath the capacity curve, and we identify a proper maze-like structure so that each passage of the maze is crossed by only O(1) tasks in the computed solution. In combination with the known PTAS for delta-small tasks, our result implies a 2+eps approximation for UFP, improving on the previous best 7+eps approximation [Bonsma et al., FOCS 2011]. We remark that our improved approximation factor matches the best known approximation ratio for the considerably easier special case of uniform edge capacities.

cs.DS↗
arXiv2011

On Multiple Round Sponsored Search Auctions with Budgets

In a sponsored search auction the advertisement slots on a search result page are generally ordered by click-through rate. Bidders have a valuation, which is usually assumed to be linear in the click-through rate, a budget constraint, and receive at most one slot per search result page (round). We study multi-round sponsored search auctions, where the different rounds are linked through the budget constraints of the bidders and the valuation of a bidder for all rounds is the sum of the valuations for the individual rounds. All mechanisms published so far either study one-round sponsored search auctions or the setting where every round has only one slot and all slots have the same click-through rate, which is identical to a multi-item auction. This paper contains the following three results: (1) We give the first mechanism for the multi-round sponsored search problem where different slots have different click-through rates. Our mechanism is incentive compatible in expectation, individually rational in expectation, Pareto optimal in expectation, and also ex-post Pareto optimal for each realized outcome. (2) Additionally we study the combinatorial setting, where each bidder is only interested in a subset of the rounds. We give a deterministic, incentive compatible, individually rational, and Pareto optimal mechanism for the setting where all slots have the same click-through rate. (3) We present an impossibility result for auctions where bidders have diminishing marginal valuations. Specifically, we show that even for the multi-unit (one slot per round) setting there is no incentive compatible, individually rational, and Pareto optimal mechanism for private diminishing marginal valuations and public budgets.

cs.GT↗
arXiv2011

Approximation Algorithms for Union and Intersection Covering Problems

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in a union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, intersection k-MST can formalize the problem of connecting a subset of users to both electricity and water. We present a number of hardness and approximation results for union and intersection versions of several standard optimization problems: MST, Steiner tree, set cover, facility location, TSP, and their partial covering variants.

cs.DS↗
arXiv2010

Combinatorial Auctions with Budgets

We consider budget constrained combinatorial auctions where bidder $i$ has a private value $v_i$, a budget $b_i$, and is interested in all the items in $S_i$. The value to agent $i$ of a set of items $R$ is $|R \cap S_i| \cdot v_i$. Such auctions capture adword auctions, where advertisers offer a bid for ads in response to an advertiser-dependent set of adwords, and advertisers have budgets. It is known that even of all items are identical and all budgets are public it is not possible to be truthful and efficient. Our main result is a novel auction that runs in polynomial time, is incentive compatible, and ensures Pareto-optimality for such auctions when the valuations are private and the budgets are public knowledge. This extends the result of Dobzinski et al. (FOCS 2008) for auctions of multiple {\sl identical} items and public budgets to single-valued {\sl combinatorial} auctions with public budgets.

cs.GT↗