Operations Research Letters 36 (2008) 726–733 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl A note on the continuous mixing set M. Zhao a , I.R. de Farias Jr. b,∗ a IBM T.J. Watson Research, United States b Department of Industrial Engineering, Texas Tech University, United States article info Article history: Received 15 October 2007 Accepted 4 August 2008 Available online 15 August 2008 Keywords: Mixed-integer programming Branch-and-cut Polyhedral combinatorics Simple mixed-integer set abstract The continuous mixing set is S =  (s, r , z ) ∈ R × R n + × Z n : s + r j + w j z j ≥ f j , j = 1,..., n  , where w 1 ,...,w n > 0 and f 1 ,..., f n ∈ R. Let m = |{w 1 ,...,w n }|. We show that when w 1 |···|w n , optimization over S can be performed in time O(n m+1 ), and in time O(n log n) when w 1 =···= w n = 1. © 2008 Elsevier B.V. All rights reserved. 1. Introduction It is widely accepted that the overwhelming progress of computational mixed-integer programming (MIP) over the past fifteen years is largely due to the use of cutting planes in branch-and-bound algorithms, see for example Fügenschuh and Martin [8] and Wolsey [17]. Recently, an interesting direction for studying cutting planes for MIP emerged from Marchand and Wolsey’s [12] observation, and its successful use in computation, that several families of well-known cutting planes, including 0–1 cover inequalities [2,11,16] and Gomory cuts [9], are implied by the only non-trivial facet-defining inequality valid for conv(S MIR ): x + (b −⌊b⌋)z ≥ (b −⌊b⌋)⌈b⌉, where: S MIR ={(x, z ) ∈ R + × Z : x + z ≥ b} is the mixed-integer rounding (MIR) set [14]. It consists in studying simple MIP sets and their computational use. Specifically, we say that an MIP set is simple if optimization over it can be performed in polynomial time. A major goal of this line of research is to produce families of cutting planes for MIP that supersede the MIR cuts and their close relatives (i.e. Gomory [9], intersection [1], disjunctive [3], and split [4] cuts). In order to achieve this objective, it is necessary first to identify simple sets that generalize S MIR ; second to study the convex hull of such sets (specifically, determine extreme points and rays, and valid inequalities for the set in question) or polynomial size formulations that project onto the set under consideration, i.e. an extended formulation; and third, to develop strategies to use the simple set computationally (specifically, determine how to relax the MIP to a problem over the simple set and how to use the cuts valid for the simple set to separate points that are not valid for the MIP). Let n be a positive integer and N ={1,..., n}. To address these questions, Günlük and Pochet [10] introduced the mixing-MIR set S MMIR , given by: S MMIR ={(x, z ) ∈ R + × Z n : x + z j ≥ b j , j ∈ N }. They gave all extreme points, extreme rays, and a full inequality description of conv(S MMIR ), and initial computational results on the market share problem that use cuts implied by the inequalities valid for conv(S MMIR ). Since then, several sets generalizing S MMIR have appeared in the literature. A particularly interesting one is the continuous mixing set : S (w) =  (s, r , z ) ∈ R × R n + × Z n : (1) holds  , ∗ Corresponding author. E-mail addresses: mzhao@us.ibm.com (M. Zhao), ismael.de-farias@ttu.edu (I.R. de Farias Jr.). 0167-6377/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2008.08.001