Frobenius Problem and the Covering Radius of a Lattice Lenny Fukshansky and Sinai Robins
نویسنده
چکیده
Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as ∑ N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1, ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.
منابع مشابه
Frobenius Problem and the Covering Radius of a Lattice
Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as P N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1 , ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-har...
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