Online Matching Optimization via Convex Cost Flow Reformulation
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Solution Overview
Problem
Existing online matching techniques suffer from low approximation rates and prolonged calculation times, especially when dealing with large remaining amounts of nodes, and there is a need for a more efficient method to determine matching strategies and control appearance probabilities.
Innovation Solution
An optimizing apparatus that includes an acquisition unit, formulation unit, determination unit, transformation unit, and problem solving unit to transform the optimization problem into a minimum convex cost flow problem, enabling an approximate solution with a better approximation rate and reduced calculation time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If existing online matching techniques are used, then the matching strategy can be determined, but the approximation rate is low (as low as 1/2) and calculation time increases when remaining amount is large
Solution Approach 1:
The patent transforms the original online matching problem parameters by introducing a new variable y_u,v that represents the expected number of matches, and reformulates the objective function in terms of these new parameters. This parameter transformation enables the use of convex optimization techniques that achieve better approximation rates while maintaining computational efficiency even when remaining amounts are large
Solution Approach 2:
The patent introduces an intermediary formulation layer that converts the difficult-to-solve online matching problem into an equivalent convex cost flow problem. This intermediary transformation allows the use of efficient flow-based algorithms that provide both theoretical approximation guarantees and practical computational speed, resolving the contradiction between solution quality and calculation time
2Productivity
If the remaining amount ru of each node u takes a large value, then more matches can be made, but calculation time becomes prolonged
Solution Approach 1:
The patent replaces the traditional mechanical optimization approach with convex optimization and flow-based algorithms. By substituting the solution methodology rather than merely optimizing parameters, the system achieves efficient computation even for large remaining amounts, as the convex formulation allows for polynomial-time solutions that scale well with problem size
3Measurement precision
If the approximation rate is improved, then better solution quality is achieved, but the device complexity or algorithm complexity increases
Solution Approach 1:
The patent segments the complex online matching problem into two more manageable components: a convex optimization subproblem for determining optimal weights and expected matches, and a greedy matching subproblem for actual allocation. This segmentation allows each component to be solved efficiently with appropriate algorithms, achieving better approximation rates without excessive overall complexity
Data Source
AI summary
A optimizing apparatus capable of solving an online matching problem according to an embodiment includes: circuitry configured to acquire input data including information regarding the node, a remaining amount given to a fixed node among nodes, the appearance probability given to an appearance node among the nodes, and a reward given to each edge when matching is performed, perform formulation to a first optimization problem based on the input data, determine whether or not all of the appearance nodes satisfy a predetermined assumption, perform transformation into a second optimization problem capable of obtaining an approximate solution that is a variable that controls a weight of each node and the appearance probability in the first optimization problem and a matching strategy in a case where the predetermined assumption is satisfied, obtain the approximate solution by solving the second optimization problem, and output the approximate solution.