Algorithms and Complexities of Matching Variants in Covariate Balancing

In an observational study there are two disjointed groups of samples, one of treatment samples and the other of control samples. Each of the samples is characterized by several observed covariates. Covariate balancing problems arise when estimating causal effects using observational data. It is desirable to replicate a randomized experiment by obtaining treatment and control groups with similar covariate distributions. Even though covariate balancing problems have been studied extensively, the complexity status of many variants has not been established. Some of our results demonstrate that 2-covariate balancing problems are polynomial time solvable, whereas almost all problems are hard for three or more covariates. These results have practical implications, such as justifying the use of implicit enumeration techniques or heuristics for the hard cases. A new approach suggested by our results is to use the 2-covariate polynomial cases and relax the problem by aggregating covariates into two sets to be solved efficiently. (accepted paper title: Algorithms and complexities of matching variants in covariate balancing) Here, we study several variants of matching problems that arise in covariate balancing. Covariate balancing problems can be viewed as variants of matching, or b-matching, with global side constraints. We present here a comprehensive complexity study of the covariate balancing problems providing polynomial time algorithms, or a proof of NP-hardness. The polynomial time algorithms described are mostly combinatorial and rely on network flow techniques. In addition, we present several fixed-parameter tractable results for problems where the number of covariates and the number of levels of each covariate are seen as a parameter. Funding: This work was supported by National Science Foundation [Grants CMMI-1760102 and NSF 2112533]; Israel Science Foundation (308/18). Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2286 .