[Paper Review] A Stochastic Probing Problem with Applications

Anupam Gupta, Viswanath Nagarajan|arXiv (Cornell University)|Jan 1, 2013

Game Theory and Voting Systems17 citations

TL;DR

This paper introduces a unified framework for stochastic probing problems under general packing constraints (e.g., k-systems, matroid intersections), proposing a greedy algorithm for unweighted cases and an LP-based rounding approach with contention resolution schemes for weighted cases. It achieves an Ω(1/(k_in + k_out))-approximation for weighted probing under k_in- and k_out-matroid constraints, enabling the first polynomial-time sequential posted-price mechanisms with constant-factor approximation under k-matroid intersection feasibility.

ABSTRACT

We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights {w_e} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p_e values-- to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation: (1) the set Q of probed elements satisfy an "outer" packing constraint, and (2) the set S of chosen elements satisfy an "inner" packing constraint. The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching and Bayesian mechanism design, and can also handle more general constraints. As an application, we obtain the first polynomial-time $Ω(1/k)$-approximate "Sequential Posted Price Mechanism" under k-matroid intersection feasibility constraints.

Motivation & Objective

• To develop a general framework for stochastic probing under complex packing constraints, including intersections of matroids and knapsacks.
• To unify and generalize prior results in stochastic matching and Bayesian mechanism design under a single model.
• To provide a polynomial-time approximation algorithm for sequential posted-price mechanisms under k-matroid intersection feasibility constraints.
• To extend the framework to include global time deadlines in probing, modeling real-world constraints like patient timeouts in kidney exchange.
• To establish tight approximation guarantees using LP relaxation and contention resolution schemes for weighted and unweighted cases.

Proposed method

• Propose a stochastic probing model where elements are probed to reveal their active status, with active elements irrevocably added to the solution set.
• Enforce two downward-closed packing constraints: an inner constraint (I_in) on the chosen set and an outer constraint (I_out) on the probed set.
• For unweighted probing, apply a greedy algorithm that probes elements in decreasing order of activation probability pe, respecting both I_in and I_out constraints.
• For weighted probing, use an LP relaxation with variables representing marginal probabilities of probing and choosing each element.
• Apply contention resolution (CR) schemes to round the fractional LP solution, ensuring feasibility under I_in and I_out.
• For deadline extensions, model time constraints as a laminar matroid L and relate the deadline problem to a relaxed probing instance with I_out ∩ L as the outer constraint.

Experimental results

Research questions

• RQ1 Can a single, unified framework be developed to model and solve stochastic probing problems under general packing constraints, including matroid and knapsack intersections?
• RQ2 What approximation ratio can be achieved by a greedy algorithm for unweighted stochastic probing when both inner and outer constraints are k-systems?
• RQ3 How can LP-based rounding with contention resolution schemes be used to achieve constant-factor approximation for weighted stochastic probing under k-matroid constraints?
• RQ4 Can the framework be extended to handle global time deadlines, such as in kidney exchange with patient time limits?
• RQ5 What is the approximation guarantee for sequential posted-price mechanisms under k-matroid intersection feasibility constraints?

Key findings

• The greedy algorithm for unweighted stochastic probing achieves a tight 1/(k_in + k_out)-approximation ratio when the inner and outer constraints are k_in- and k_out-systems, respectively.
• For weighted stochastic probing under intersections of k_in and k_out matroids, the paper achieves an Ω(1/(k_in + k_out))-approximation using LP relaxation and contention resolution schemes.
• Under arbitrary k_in- and k_out-system constraints, the approximation ratio degrades to Ω(1/(k_in + k_out)^2), which remains constant for fixed k.
• The framework enables the first polynomial-time Ω(1/k)-approximate sequential posted-price mechanism under k-matroid intersection feasibility constraints, improving prior work.
• For unweighted probing with deadlines, a greedy algorithm achieves a 1/(2(k_in + k_out + 1))-approximation, with a novel coupling argument using laminar matroid constraints.
• The approach generalizes and unifies results in stochastic matching and Bayesian mechanism design, providing a simpler, LP-based proof for the 4-approximation in unweighted stochastic matching.