The Subset Sum Matching Problem

Abstract: This paper presents a new combinatorial optimisation task, the Subset Sum Matching Problem (SSMP), which is an abstraction of common financial applications such as trades reconciliation. We present three algorithms, two suboptimal and one optimal, to solve this problem. We also generate a benchmark to cover different instances of SSMP varying in complexity, and carry out an experimental evaluation to assess the performance of the approaches.

• The paper introduces the SSMP, a novel combinatorial optimization problem for financial data reconciliation with a clear formal framework and matching criteria.
• It proposes three algorithms—an optimal MILP solver and two sub-optimal methods based on search and dynamic programming—to address the matching challenge.
• Experimental results demonstrate trade-offs between optimality and efficiency, with the dynamic programming approach excelling in real-valued, large-scale problem scenarios.

The Subset Sum Matching Problem: A Comprehensive Analysis

The Subset Sum Matching Problem (SSMP) represents a novel combinatorial optimization problem inspired by practical applications within the domain of financial data reconciliation. Introduced by Wu et al., this problem is framed within the context of matching elements from two multisets under specific conditions, aimed at optimizing a defined objective (2508.19218). This essay explores the problem's definition, its relation to other combinatorial optimization problems, the proposed algorithms for solving it, and experimental results assessing these algorithms' performances.

Problem Definition and Theoretical Foundations

SSMP formalizes a subset of combinatorial optimization problems where given two multisets of real-valued objects, the objective is to find pairs of subsets whose element sums are within a predefined tolerance $\epsilon$. This is achieved by maximizing a solution quality measure $\Psi(s)$, which encourages forming multiple matches with extensive coverage of elements.

The problem builds on the Subset Matching Problem (SMP), where defining validity is contingent on a Boolean function applied to subsets of the input multisets (Figure 1).

Figure 1: An example of SSMP with input multisets $a, b$, tolerance $\epsilon$, and three feasible solutions $s_1, s_2, s_3$.

SSMP is akin to classical problems like the subset sum and hypergraph matching problems, borrowing conceptual elements while extending upon them, especially in terms of flexibility regarding the function defining subset validity.

Algorithmic Solutions

Three primary algorithms are introduced to tackle SSMP: an optimal solver using mixed-integer linear programming (MILP) and two sub-optimal solvers based on search and dynamic programming (DP). The MILP approach ensures optimal solutions by formulating SSMP as a linear program, efficiently managing complexities like match inclusion and objective optimization.

The search-based solver leverages a combinatorial schema, maintaining a cache of subset sums for efficient match identification, iterating over possible subsets from one multiset and seeking counterparts within a pre-computed table derived from the other multiset. Alternatively, the DP approach constructs a tabular representation of potential subset sums, utilizing tree search methodologies to validate and retrieve feasible matches (Figure 2).

Figure 2: Illustration of the search algorithm for solving an $\text{SSMP}^-$ instance with pre-calculation and matching steps.

The search and DP solvers eschew optimal guarantees for efficiency when operating under constraints common in real-world datasets, making them particularly suited for applications where rapid approximate solutions are desirable (Figure 3).

Figure 3: Illustration of the DP approach showing discretization, tabulation, and tree search phases.

Empirical Evaluation

Experimental evaluations reveal insights into each algorithm's strengths and nuances across varying problem scales and complexities. Integer instances showcase the MILP solver's robust solution capability albeit with time limitations in large-scale problems, whereas the search and DP solvers demonstrate more consistent performance across configurations, with the DP solver particularly excelling in scenarios with realistic value bounds due to better time complexity.

For real-valued instances, considerations like discretization become crucial. Here, the DP solver's superior handling of large problem instances highlights its practical utility in non-integer scenarios, offering efficient matching within constrained computational resources.

Conclusion

The introduction of SSMP underscores a significant advancement in combinatorial optimization tailored to financial reconciliation tasks. While the optimal MILP-based approach offers analytical rigour, the sub-optimal, heuristics-driven algorithms provide necessary agility for practical deployments. Future research could further refine these methodologies, explore approximation strategies with tighter bounds, or apply SSMP principles to other domains necessitating matched data validation under uncertainty.

Exploration across broader applications or enhancements incorporating machine learning for dynamic run-time adaptations could propel SSMP to address even more complex real-world challenges, maintaining theoretical integrity while expanding practical relevance.