[Paper Review] Faster Fundamental Graph Algorithms via Learned Predictions

Justin Y. Chen, Sandeep Silwal+2 more|arXiv (Cornell University)|Apr 26, 2022

Machine Learning and Algorithms4 citations

TL;DR

This paper presents learning-augmented algorithms for fundamental graph problems using predicted dual variables, achieving faster runtimes by leveraging machine-learned hints. It introduces a novel use of reduced edge duals for shortest paths and a general reduction framework to extend learning-augmented algorithms to new problems, with provable learnability guarantees for the predictions via PAC learning theory.

ABSTRACT

We consider the question of speeding up classic graph algorithms with machine-learned predictions. In this model, algorithms are furnished with extra advice learned from past or similar instances. Given the additional information, we aim to improve upon the traditional worst-case run-time guarantees. Our contributions are the following: (i) We give a faster algorithm for minimum-weight bipartite matching via learned duals, improving the recent result by Dinitz, Im, Lavastida, Moseley and Vassilvitskii (NeurIPS, 2021); (ii) We extend the learned dual approach to the single-source shortest path problem (with negative edge lengths), achieving an almost linear runtime given sufficiently accurate predictions which improves upon the classic fastest algorithm due to Goldberg (SIAM J. Comput., 1995); (iii) We provide a general reduction-based framework for learning-based graph algorithms, leading to new algorithms for degree-constrained subgraph and minimum-cost $0$-$1$ flow, based on reductions to bipartite matching and the shortest path problem. Finally, we give a set of general learnability theorems, showing that the predictions required by our algorithms can be efficiently learned in a PAC fashion.

Motivation & Objective

• To accelerate classic graph algorithms using machine-learned predictions, especially dual variables from prior or similar instances.
• To address whether learning-augmented algorithms can be generalized beyond bipartite matching without redesigning from scratch.
• To develop a general reduction-based framework that enables transfer of learning-augmented algorithms to new graph problems.
• To establish theoretical foundations for the learnability of required predictions using PAC learning.

Proposed method

• Introduces reduced edge duals as a new type of prediction for shortest path problems with negative weights, enabling faster convergence.
• Proposes a rounding scheme to convert predicted reduced edge duals into feasible dual solutions for shortest paths.
• Develops a general reduction framework that maps new graph problems (e.g., degree-constrained subgraph, unit-capacity flow) to known problems (bipartite matching, shortest paths) with learned predictions.
• Employs empirical risk minimization over prediction error in ℓ₁ and ℓ∞ norms to learn dual predictions efficiently.
• Applies PAC learning theory to bound sample complexity and generalization error for the prediction functions.
• Uses computational complexity bounds (via VC and pseudo-dimension) to establish learnability of prediction functions based on arithmetic operations.

Experimental results

Research questions

• RQ1 Can learning-augmented algorithms be designed for fundamental graph problems beyond bipartite matching, such as single-source shortest paths with negative weights?
• RQ2 Is it possible to avoid designing a new learning-augmented algorithm for each graph problem by reusing existing ones through reductions?
• RQ3 What conditions ensure that the required predictions (e.g., dual variables) are efficiently learnable in a PAC sense?
• RQ4 How do different prediction norms (ℓ₁, ℓ∞) affect the runtime and learnability of learning-augmented graph algorithms?
• RQ5 Can the computational complexity of prediction functions be used to derive general learnability bounds?

Key findings

• The paper achieves an improved runtime of O(m√n + (m+n log n) ||y* - ŷ||₀) for minimum-weight bipartite matching, significantly improving over the prior Õ(min{m√n ||y* - ŷ'||₁, mn}) bound.
• For single-source shortest paths with negative edge weights, the algorithm runs in O(m min{||ŷ - y*||₁ · ||ŷ - y*||∞, √n log ||ŷ - y*||∞}) time when predictions are accurate.
• The reduction framework enables new learning-augmented algorithms for degree-constrained subgraph and unit-capacity minimum-cost 0-1 flow by reducing to bipartite matching and shortest paths.
• The paper proves that predictions can be learned in a PAC fashion with sample complexity bounded by O(poly(T(m,d))) for prediction functions computable in T(m,d) time.
• For ℓ₁ and ℓ∞ prediction losses, the sample complexity is bounded using pseudo-dimension and VC dimension, with the latter tied to the complexity of computing the prediction function.
• The framework establishes that predictions based on edge features and linear functions are efficiently learnable under standard assumptions, with generalization error bounds derived from computational complexity.