ELSEVIER European Journal of Operational Research 101 (1997) 74-80 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Greedy sets and related problems Eberhard Girlich a,,, Michail M. Kovalev b, Dmitri M. Vasilkov b,1 a Fakultiit fu'r Mathematik, Otto-von-Guericke- Universitiit Magdeburg, PSF 4120, 39016 Magdeburg, Germany b Faculty of Applied Mathematics and lnformatics, Belarus State University, E Skaryna 4, 220050 Minsk. Belarus Received 5 January 1996; accepted 3 September 1996 Abstract A set D is called greedy if the greedy algorithm solves the problem max{cx [ x E D} for any c such that c~ >/ --- /> c, /> 0. In previous articles we described the class of all greedy convex polyhedra in/R n using the partial order technique. Here we generalize the problem and extend the class of greedy feasible sets using the so-called generalized canonical order. These results are spread onto some related problems (characterization of greedy matrices and functions). (~ 1997 Published by Elsevier Science B.V. Keywords: Greedy algorithm; Canonical order; Greedy matrix; Greedy functions 1. Introduction The greedy algorithm is a fundamental optimiza- tion procedure which provides a considerable interest for investigators due to its two features: simple imple- mentation and good speed. In general, the greedy algo- rithm constructs a feasible solution xg for the problem Max f(x) (1) s.t. xED by the following scheme. • Ordering variables according to a permutation 7r = {rr(1) ..... 7r(n)}. The choice of the ordering rule depends on the problem to be solved. • Computing variables by x~(1) =max{x~(l) I x C D} * Corresponding author. I Supported by the Belarusian Fundamental Research Fund, DAAD and BMBWFT (grant 03-ZI7BRI). and xg~(k) = max{ x,~(k) I x E D, g i=I,. k-I}, Xrr(i) = Xlr(i ) , . • , for k=2 ..... n. Thus, the greedy solution xg is defined as the lex- icographical maximum of the feasible set. Moreover, if D is a bounded polyhedron in R n, then xg is a basic solution. The general problem is to determine under what conditions xg is optimal. This general problem has a lot of interpretations de- pending on what we mean with the word 'conditions'. Here we distinguish the following five classes of re- lated problems. 1) Consider a linear problem Max cx s.t. x E D = {x l Ax <~ b, O <<.x <<. d}. (2) 0377-2217/97/$17.00 (~) 1997 Published by Elsevier Science B.V. All rights reserved. PHS0377-2217(96)00270-6