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In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted-price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed-price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most k distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most k thresholds. We present asymptotic results in k and results for small values of k. For k = 2 prices, we show an improvement of at least 11% over the best fixed-price solution. Moreover, k = 5 prices suffice to guarantee almost 99% of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems. 

Motivated by fairness concerns, we study existence and computation of portfolios, defined as: given anoptimization problem with feasible solutions D, a class C of fairness objective functions, a set X of feasible solutions is an α-approximate portfolio if for each objective f in C, there is an α-approximation for f in X. We study the trade-off between the size |X| of the portfolio and its approximation factor αfor various combinatorial problems, such as scheduling, covering, and facility location, and choices of C as top-k, ordered and symmetric monotonic norms. Our results include: (i) an α-approximate portfolio of size O(log d*log(α/4))for ordered norms and lower bounds of size \Omega( log dlog α+log log d)for the problem of scheduling identical jobs on d unidentical machines, (ii) O(log n)-approximate O(log n)-sized portfolios for facility location on n points for symmetric monotonic norms, and (iii) logO(d)^(r^2)-size O(1)-approximate portfolios for ordered norms and O(log d)-approximate for symmetric monotonic norms for covering polyhedra with a constant rnumber of constraints. The latter result uses our novel OrderAndCount framework that obtains an exponentialimprovement in portfolio sizes compared to current state-of-the-art, which may be of independent interest. 

In the minimum eigenvalue problem, we are given a collection of vectors and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix formed by the sum of their outer products. We give a -time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least $$1-\epsilon$$ times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem. 

Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far. Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged. 

Given a set of facilities and clients, and costs to open facilities, the classic facility location problem seeks to open a set of facilities and assign each client to one open facility to minimize the cost of opening the chosen facilities and the total distance of the clients to their assigned open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms. In this work, we consider a fair version of the problem where we are given 𝑟 clients groups that partition the set of clients, and the distance of a given group is defined as the average distance of clients in the group to their respective open facilities. The objective is to minimize the Minkowski 𝑝-norm of vector of group distances, to penalize high access costs to open facilities across 𝑟 groups of clients. This generalizes classic facility location (𝑝 = 1) and the minimization of the maximum group distance (𝑝 = ∞). However, in practice, fairness criteria may not be explicit or even known to a decision maker, and it is often unclear how to select a specific "𝑝" to model the cost of unfairness. To get around this, we study the notion of solution portfolios where for a fixed problem instance, we seek a small portfolio of solutions such that for any Minkowski norm 𝑝, one of these solutions is an 𝑂(1)-approximation. Using the geometric relationship between various 𝑝-norms, we show the existence of a portfolio of cardinality 𝑂(log 𝑟), and a lower bound of (\sqrt{log r}). There may not be common structure across different solutions in this portfolio, which can make planning difficult if the notion of fairness changes over time or if the budget to open facilities is disbursed over time. For example, small changes in 𝑝 could lead to a completely different set of open facilities in the portfolio. Inspired by this, we introduce the notion of refinement, which is a family of solutions for each 𝑝-norm satisfying a combinatorial property. This property requires that (1) the set of facilities open for a higher 𝑝-norm must be a subset of the facilities open for a lower 𝑝-norm, and (2) all clients assigned to an open facility for a lower 𝑝-norm must be assigned to the same open facility for any higher 𝑝-norm. A refinement is 𝛼-approximate if the solution for each 𝑝-norm problem is an 𝛼-approximation for it. We show that it is sufficient to consider only 𝑂(log 𝑟) norms instead of all 𝑝-norms, 𝑝 ∈ [1, ∞] to construct refinements. A natural greedy algorithm for the problem gives a poly(𝑟)-approximate refinement, which we improve to poly(r^1/\sqrt{log 𝑟})-approximate using a recursive algorithm. We improve this ratio to 𝑂(log 𝑟) for the special case of tree metric for uniform facility open cost. Our recursive algorithm extends to other settings, including to a hierarchical facility location problem that models facility location problems at several levels, such as public works departments and schools. A full version of this paper can be found at https://arxiv.org/abs/2211.14873. 

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its “combinatorial shadow”. We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities.