Network Equilibrium Pricing via Combinatorial Auction Algorithms
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Solution Overview
Problem
Current methods for finding equilibria in natural resource allocation markets are inefficient, relying on slow and resource-intensive convex programs, and there is a lack of combinatorial solvability for resource allocation markets with equilibria captured via convex programs similar to the Eisenberg-Gale program.
Innovation Solution
The development of strongly polynomial combinatorial algorithms to determine equilibria in natural resource allocation markets, allowing for price adjustments to converge to equilibrium in polynomial time, particularly in computer networks by analyzing edge utilization and adjusting prices to balance demand and supply.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If convex programs are used to find equilibrium solutions, then equilibrium can be found, but the computation is very slow and resource intensive
Solution Approach 1:
The patent replaces the continuous convex optimization approach with a discrete combinatorial algorithm. Specifically, it uses a primal-dual combinatorial algorithm that operates on network flows and matchings rather than continuous variables, achieving polynomial time complexity while maintaining equilibrium solution accuracy. This substitution transforms the computational mechanism from continuous optimization to discrete combinatorial operations.
Solution Approach 2:
The patent changes the parameter representation from continuous utility values to discrete price vectors and flow allocations. By representing equilibrium conditions through discrete price vectors p and flow vectors f that satisfy complementary slackness conditions, the algorithm achieves polynomial-time solvability while preserving the equilibrium properties that require continuous parameter precision.
2Reliability
If continuous time algorithms are used for congestion control, then equilibrium can be approached, but there are no time-efficiency guarantees
Solution Approach 1:
The patent performs preliminary action by directly computing the equilibrium price vector and flow allocation through combinatorial operations rather than gradually approaching it through continuous time dynamics. The algorithm initializes price vectors and iteratively adjusts them using combinatorial updates based on residual demands and supplies, achieving equilibrium in polynomial steps without requiring continuous time simulation.
Solution Approach 2:
The patent substitutes continuous time differential equations with discrete combinatorial iteration. Instead of using continuous congestion control dynamics that lack time guarantees, the algorithm uses discrete price adjustment rules based on combinatorial optimization principles, providing explicit polynomial time complexity bounds for reaching equilibrium.
3Productivity
If combinatorial algorithms are designed for linear programs, then efficiency is improved, but they cannot solve nonlinear programs like Eisenberg-Gale convex program
Solution Approach 1:
The patent changes the parameter representation by introducing price vectors as dual variables that transform the nonlinear Eisenberg-Gale objective into a combinatorial pricing problem. By representing the nonlinear utility maximization through linear pricing constraints and complementary slackness conditions, the algorithm extends combinatorial methods to nonlinear programs while maintaining polynomial time complexity.
Solution Approach 2:
The patent introduces price vectors as an intermediary between the nonlinear utility functions and the combinatorial optimization framework. These price vectors serve as mediators that translate nonlinear equilibrium conditions into linear pricing constraints that can be solved using combinatorial algorithms, thereby bridging the gap between nonlinear program theory and combinatorial algorithm practice.
Data Source
AI summary
The claimed subject matter provides an architecture for facilitating equilibrium solutions for resource allocation markets. One particular example of such markets can be a computer network environment. The architecture can model buyers as nodes and suppliers as edges, incorporating notions of an ascending price auction in order to provide optimal equilibrium solution to Eisenberg-Gale type convex programs in strongly polynomial time.