arXiv subjects

Zohar Barak

Publications and source records attributed to Zohar Barak.

5 recordsLinked to original sources

arXiv2026

Ex-ante versus Ex-post: Egalitarian Facility Location Mechanism Design

We study the facility location mechanism design problem where $n$ strategic agents report locations in Euclidean space and the mechanism outputs a single facility location. Each agent's cost is its distance from the facility, and our objective is to minimize the egalitarian cost, i.e., the maximum agent cost, in a strategyproof way. The optimal deterministic approximation ratio is $2$, achieved by any dictator mechanism. We study the power of randomized strategyproof-in-expectation mechanisms. Prior work has focused on ex-post evaluation, defined as the expected maximum agent cost. We instead study ex-ante evaluation, defined as the maximum expected agent cost, which is naturally aligned with strategyproofness in expectation. We establish the following results: (1) Low dimensions: Strict ex-ante vs. ex-post separation. In $\mathbb{R}$, we give a simple strategyproof mechanism achieving the optimal ex-ante approximation ratio of $1$. In $\mathbb{R}^2$, we design the "Random Rotated Corner" mechanism, with ex-ante approximation ratio at most $1.598$, breaking the deterministic barrier. For the ex-post objective, we prove a lower bound of $1.605$, yielding a strict separation in $\mathbb{R}^2$. (2) High dimensions: Impossibility. In $\mathbb{R}^d$ for $d \gg 1$, we show that no strategyproof-in-expectation mechanism improves on the deterministic dictator mechanism beyond $o_d(1)$. Thus neither ex-post nor ex-ante evaluation yields improved fairness guarantees in high dimensions. An implication is that the "Random Rotation Coordinate-Wise Median" (RRCWM), currently the best known mechanism for the utilitarian objective, is also best possible for the egalitarian objective in high dimension: we show it achieves an approximation ratio of $2$ for both ex-post and ex-ante objectives in $\mathbb{R}^d$ for every $d \ge 1$.

cs.GT↗

arXiv2026

Facility Location Mechanism Design: Breaking The Deterministic Barrier

We study the facility location mechanism design problem where $n$ agents report their locations in Euclidean space, and the output is a single facility location. The cost function of each agent is the distance from the returned facility, and the objective is to minimize the social cost function (the sum of agent costs) in a strategyproof way. Our contributions: 1. Breaking the deterministic barrier. For $\mathbb{R}^2$, we give a random strategyproof mechanism (RR-CWM) achieving an expected approximation ratio of $\frac{4}{\pi} \approx 1.27$, which strictly improves upon the best deterministic strategyproof mechanism (which has a $\sqrt{2} \approx 1.41$ ratio). This closes the open problem of separating deterministic and random mechanisms for utilitarian facility location mechanism design in $\mathbb{R}^2$. For $\mathbb{R}^d$, we show that the expected approximation ratio of our mechanism is in $[1.41 - O(1/\sqrt{d}), 1.547]$. 2. Improved learning augmented mechanisms through randomization. We show our ideas can achieve better performance in the learning augmented setting in $\mathbb{R}^2$, where in addition to the input the mechanism also receives predictions. For the output prediction model of Agrawal et al. 2022 we show an improved expected consistency-robustness trade-off. Our results also imply improved performance for the input MAC predictions model of Barak et al. 2024. 3. The limitations of Random Dictators. We show a lower bound for the common mechanism class of GRD (Generalized Random Dictator) mechanisms, where only locations reported by the agents may be returned. We show that any GRD mechanism has a larger expected approximation ratio than our RR-CWM mechanism, as our lower bound for $\mathbb{R}^2$ is $\frac{4}{\pi}$ (matching the upper bound of RR-CWM, which is not a GRD mechanism). For $\mathbb{R}^d$, we show a lower bound of $\sqrt{2} - O(1/d)$.

cs.GT↗

arXiv2026

Stochastic Knapsack: Semi-Adaptivity Gaps and Improved Approximation

In stochastic combinatorial optimization, algorithms differ in their adaptivity: whether or not they query realized randomness and adapt to it. Dean et al. (FOCS '04) formalize the adaptivity gap, which compares the performance of fully adaptive policies to that of non-adaptive ones. We revisit the fundamental Stochastic Knapsack problem of Dean et al., where items have deterministic values and independent stochastic sizes. A policy packs items sequentially, stopping at the first knapsack overflow or before. We focus on the challenging risky variant, in which an overflow forfeits all accumulated value, and study the problem through the lens of semi-adaptivity: We measure the power of $k$ adaptive queries for constant $k$ through the notions of $0$-$k$ semi-adaptivity gap (the gap between $k$-semi-adaptive and non-adaptive policies), and $k$-$n$ semi-adaptivity gap (between fully adaptive and $k$-semi-adaptive policies). Our first contribution is to improve the classic results of Dean et al. by giving tighter upper and lower bounds on the adaptivity gap. Our second contribution is a smoother interpolation between non-adaptive and fully-adaptive policies, with the rationale that when full adaptivity is unrealistic (due to its complexity or query cost), limited adaptivity may be a desirable middle ground. We quantify the $1$-$n$ and $k$-$n$ semi-adaptivity gaps, showing how well $k$ queries approximate the fully-adaptive policy. We complement these bounds by quantifying the $0$-$1$ semi-adaptivity gap, i.e., the improvement from investing in a single query over no adaptivity. As part of our analysis, we develop a 3-step "Simplify-Equalize-Optimize" approach to analyzing adaptive decision trees, with possible applications to the study of semi-adaptivity in additional stochastic combinatorial optimization problems.

cs.DS↗

arXiv2025

Approximating Pandora's Knapsack via Simple Policies

We introduce Pandora's Knapsack: a hybrid between the classic stochastic knapsack problem [Dean et al., 2008] and Pandora's box [Weitzman, 1979]. As in stochastic knapsack, items have sizes drawn from known distributions, and items that fit within a knapsack contribute to the total value. As in Pandora, every item $i$ comes in a box that costs $c_i$ to open. What distinguishes our problem is that the size is revealed only after opening the box and paying the cost. We study the power of simple decision-making policies to approximate Pandora's Knapsack, along two complexity axes: (i)~knowledge of size distributions, and (ii)~adaptivity. We show that adding costs to stochastic knapsack completely changes the algorithmic landscape, and policies must now be complex along both axes to be approximately-optimal. We complement these impossibilities by showing that with full distributional information and slightly more adaptivity---allowing adaptive skipping of items---a constant approximation can be recovered. Our analysis reveals an economic quantity, namely ROI (return-on-investment, defined as the jobs' minimum utility over cost), which smoothly characterizes the performance of simple policies. Across a hierarchy of increasingly adaptive policies, we establish near-tight adaptivity gaps all governed by ROI. To demonstrate the importance of the ROI parameter in characterizing simple policies, we revisit Pandora's Box, and show that return-on-investment exactly captures the adaptivity gap in this classic problem as well. As a corollary, we get that simple policies are approximately-optimal provided the ROI is sufficiently large.

cs.GT↗

arXiv2024

MAC Advice for Facility Location Mechanism Design

Algorithms with predictions have attracted much attention in the last years across various domains, including variants of facility location, as a way to surpass traditional worst-case analyses. We study the $k$-facility location mechanism design problem, where the $n$ agents are strategic and might misreport their location. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. However, these predictions are only "mostly" and "approximately" correct (or MAC for short) -- i.e., some $δ$-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are allowed to be correct up to an $\varepsilon$-error. We make no assumption on the independence of the errors. Can such predictions allow us to beat the current best bounds for strategyproof facility location? We show that the $1$-median (geometric median) of a set of points is naturally robust under corruptions, which leads to an algorithm for single-facility location with MAC predictions. We extend the robustness result to a "balanced" variant of the $k$ facilities case. Without balancedness, we show that robustness completely breaks down, even for the setting of $k=2$ facilities on a line. For this "unbalanced" setting, we devise a truthful random mechanism that outperforms the best known result of Lu et al. [2010], which does not use predictions. En route, we introduce the problem of "second" facility location (when the first facility's location is already fixed). Our findings on the robustness of the $1$-median and more generally $k$-medians may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.

cs.GT↗