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Observational data are prevalent in many fields of research, and it is desirable to use this data to make causal inferences. Because this data is nonrandom, additional assumptions are needed in order to construct unbiased estimators for causal effects. The standard assumption is strong ignorability, though it is often impossible to achieve the level of covariate balance that it requires. As such, researchers often settle for lesser balance levels within their datasets. However, these balance levels are generally insufficient to guarantee an unbiased estimate of the treatment effect without further assumptions. This article presents several extensions to the strong ignorability assumption that address this issue. Under these additional assumptions, specific levels of covariate balance are both necessary and sufficient for the treatment effect estimate to be unbiased. There is a trade-off, however: as balance decreases, stronger assumptions are required to guarantee estimator unbiasedness. These results unify parametric and nonparametric adjustment methods for causal inference and are actualized by the Balance Optimization Subset Selection framework, which identifies the best level of balance that can be achieved within a dataset.© 2017 Wiley Periodicals, Inc. Naval Research Logistics, 2017

The cyclic best-first search (CBFS) strategy is a recent search strategy that has been successfully applied to branch-and-bound algorithms in a number of different settings. CBFS is a modification of best-first search (BFS) that places search tree subproblems into contours which are collections of subproblems grouped in some way, and repeatedly cycles through all non-empty contours, selecting one subproblem to explore from each. In this article, the theoretical properties of CBFS are analyzed for the first time. CBFS is proved to be a generalization of all other search strategies by using a contour definition that explores the same sequence of subproblems as any other search strategy. Further, a bound is proved between the number of subproblems explored by BFS and the number of children generated by CBFS, given a fixed branching strategy and set of pruning rules. Finally, a discussion of heuristic contour-labeling functions is provided, and proof-of-concept computational results for mixed-integer programming problems from the MIPLIB 2010 database are shown. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 64–82, 2017