Claim verification in real-world settings (e.g. against a large collection of candidate evidences retrieved from the web) typically requires identifying and aggregating a complete set of evidence pieces that collectively provide full support to the claim. The problem becomes particularly challenging when there exists distinct sets of evidence that could be used to verify the claim from different perspectives. In this paper, we formally define and study the problem of identifying such minimal evidence groups (MEGs) for claim verification. We show that MEG identification can be reduced from Set Cover problem, based on entailment inference of whether a given evidence group provides full/partial support to a claim. Our proposed approach achieves 18.4% and 34.8% absolute improvements on the WiCE and SciFact datasets over LLM prompting. Finally, we demonstrate the benefits of MEGs in downstream applications such as claim generation.