saqarTvelos mecnierebaTa erovnuli akademiis moambe , t. 4, # 2, 2010 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol. 4, no. 2, 2010 © 2010 Bull. Georg. Natl. Acad. Sci. Mathematics Greedy Algorithm Fails in Compact Vector Summation George Chelidze * , Sergei Chobanyan * , George Giorgobiani * , Vakhtang Kvaratskhelia * * N. Muskhelishvili Institute of Computational Mathematics, Tbilisi (Presented by Academy Member Nikoloz Vakhania) ABSTRACT. We show that in any two-dimensional normed space there exists a collection of vectors 1 2 , , , , 1, n x x x n ≥ … such that the greedy algorithm for estimation of () 1 1 min max k i k n i x π π ≤≤ = ∑ fails to be optimal. © 2010 Bull. Georg. Natl. Acad. Sci. Key words: normed space, greedy algorithm, optimal algorithm. Let X be a linear normed space and let 1 2 , , , , 1, n x x x n ≥ … be a collection of vectors of X. Given a permutation :{1,..., } {1,..., } n n π → consider the number () 1 1 ( ) max k i k n i x π ϕπ ≤≤ = = ∑ . In many applications, for example in a series of problems of scheduling theory (see e.g. [1-3]), it is of importance to find a permutation optimal π for which ( ) ϕ π attains its minimum. We call such a permutation optimal. In scheduling theory the arrangement of summands corresponding to the optimal π is called the compact vector summation. Estimation of ( ) optimal ϕπ is frequently called the problem of (dynamical) compact vector summation (CVS). For their simplicity and constructiveness the so-called greedy algorithms in general seem to be effective. In our case the greedy algorithm (greedy permutation) to approach ( ) optimal ϕπ goes as follows: on step one we take from the collection 1 2 , , , n x x x … an element 1 n x having the minimum norm, on step two we take 2 n x such that 1 2 n n x x + is minimal, etc. Note that for the computational purposes this algorithm is of importance for it runs in polynomial time. Naturally the question arises as to whether the greedy algorithm is of the same order as the optimal one, i.e. there exists a constant C, dependent only on the space X, such that ( ) ( ) greedy optimal C ϕ π ϕ π ≤ . It can be shown that in the one-dimensional case the constant C equals 2. As to the multi-dimensional case, Jakub Wojtaszczyk (oral communication) has constructed the following systems of vectors in 2 l ∞ for which ( ) ( ) greedy optimal ϕ π ϕπ is not bounded.