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The problems of query containment, equivalence, and minimization are fundamental problems in the context of query processing and optimization. In their classic work [2] published in 1977, Chandra and Merlin solved the three problems for the language of conjunctive queries (CQ queries) on relational data, under the "set-semantics" assumption for query evaluation. While the results of [2] have been very influential in database research, it was recognized long ago that the set semantics does not correspond to the semantics of the standard commercial query language SQL. Alternative semantics, called bag and bag-set semantics, have been studied since 1993; Chaudhuri and Vardi in [5] outlined necessary and sufficient conditions for equivalence of CQ queries under these semantics. (The problems of containment of CQ bag and bag-set queries remain open to this day.) More recently, Cohen [7, 8] introduced a formalism for treating (generalizations of) CQ queries evaluated under each of set, bag, and bag-set semantics uniformly as special cases of the more general combined semantics. This formalism provides tools for studying broader classes of practical SQL queries, specifically important types of queries that arise in on-line analytical processing (OLAP). Cohen in [8] provides a sufficient condition for equivalence of (generalizations of) combined-semantics CQ queries, as well as sufficient and necessary equivalence conditions for several proper sublanguages of the query language of [8]. To the best of our knowledge, no results on minimization of CQ queries beyond set-semantics queries have been reported in the literature. Our goal in this paper is to continue the study of equivalence and minimization of CQ queries. We consider the problems of (i) finding minimized versions of combined-semantics CQ queries, and of (ii) determining whether two CQ queries are combined-semantics equivalent. We continue the tradition of [2, 5, 8] of studying these problems using the tool of containment between queries. We extend the containment, equivalence, and minimization results of [2] to general combined-semantics CQ queries, and show the limitations of each extension. We show that the minimization approach of [2] can be extended to general CQ queries without limitations. We also propose a necessary and sufficient condition for equivalence of queries belonging to a large natural sublanguage of combined-semantics CQ queries; this sublanguage encompasses (but is not limited to) all set, bag, and bag-set queries. Our equivalence and minimization results, as well as our general sufficient condition for containment of combined-semantics CQ queries, reduce correctly to the special cases reported in [5] for bag and bag-set semantics. Our containment and equivalence conditions also properly generalize the results of [8], provided the latter are restricted to the language of (combined-semantics) CQ queries.