[Paper Review] Polymatroid Prophet Inequalities

Paul Duetting, Robert Kleinberg|arXiv (Cornell University)|Jul 19, 2013

Auction Theory and Applications16 references4 citations

TL;DR

This paper extends prophet inequalities to polymatroid constraints, proving that an online algorithm can achieve at least half the expected optimal value of a weighted sum under submodular capacity constraints. The approach reduces the polymatroid problem to a block-structured matroid setting, enabling application of the Kleinberg-Weinberg algorithm to maintain the 1/2-approximation guarantee despite correlated weights within blocks.

ABSTRACT

Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen. The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint. In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.

Motivation & Objective

• To generalize prophet inequalities from matroid to polymatroid constraints, where feasible solutions are defined by submodular functions.
• To establish that an online decision-maker (gambler) can achieve at least half the expected optimal value of a weighted sum under polymatroid feasibility constraints.
• To bridge the gap in mechanism design by enabling simple, approximately optimal mechanisms for problems with polymatroid feasibility structures.
• To develop a reduction technique transforming polymatroid problems into equivalent block-structured matroid problems with correlated but block-independent weights.
• To prove that the Kleinberg-Weinberg algorithm remains effective under block-structured distributions, preserving the 1/2-approximation guarantee.

Proposed method

• Reduce a polymatroid problem with independent weights to a block-structured matroid problem by replicating each element according to its marginal contribution.
• Construct a block-structured matroid where each block corresponds to an element in the original polymatroid, and all elements in a block have identical weights.
• Use the fact that weights across different blocks are independent, while weights within a block are perfectly correlated, to model the transformed problem.
• Apply the Kleinberg-Weinberg algorithm to the block-structured matroid, which selects elements based on thresholds derived from expected maximum values.
• Prove that the expected performance of the online algorithm in the transformed problem matches the 1/2-approximation bound via a coupling argument and stochastic dominance.
• Leverage the structure of submodular polyhedra and the properties of sequential posted pricing to derive revenue guarantees in mechanism design applications.

Experimental results

Research questions

• RQ1 Can the 1/2-approximation guarantee of prophet inequalities be extended from matroid to polymatroid feasibility constraints?
• RQ2 Does the Kleinberg-Weinberg algorithm maintain its performance guarantee under block-structured distributions where weights are identical within blocks?
• RQ3 Can the reduction from polymatroid to block-structured matroid preserve the prophet inequality bound under independent weight assumptions?
• RQ4 What are the implications of this result for mechanism design in problems with polymatroid constraints, such as position auctions and spatial markets?
• RQ5 Is there a way to design simple, dominant-strategy incentive-compatible mechanisms with provable revenue guarantees in polymatroid single-parameter Bayesian settings?

Key findings

• The paper establishes a prophet inequality for polymatroid constraints: the online algorithm achieves at least half the expected optimal value of the weighted sum under submodular capacity constraints.
• The 1/2-approximation bound is tight, as it matches the known tightness in the single-choice prophet inequality setting.
• The reduction from polymatroid to block-structured matroid preserves the prophet inequality guarantee, enabling the use of existing algorithms in a new context.
• The Kleinberg-Weinberg algorithm, originally designed for independent weights, is shown to work under block-structured distributions with identical weights per block.
• In mechanism design, sequential posted pricing achieves at least half the optimal revenue for polymatroid single-parameter Bayesian problems with rational constraints.
• The result improves upon prior bounds for position auctions (e.g., GSP with reserves) and provides the first approximation guarantee for spatial markets under polymatroid constraints.