The bounded subset sum problem is almost everywhere randomly decidable in O(n) (Q1083370)

The subset sum problem is to decide, whether for given positive integers b, \(a_ i\), \(1\leq i\leq n\) the equation \[ (SS)\quad b=\max \{\sum a_ ix_ i| \sum a_ ix_ i\leq b,\quad x_ i\in \{0,1\},\quad 1\leq i\leq n\} \] is true. Given a sequence \(A=(A_ n)_{n\in N}\) of positive integers with \(A_ n=o(n)\) the A-bounded subset sum problem SS(A) is the set of all problem instances of type (SS) with \(a_ i\in \{1,...,A_ n\}\) and \(b\in \{1,...,nA_ n\}\). We show that for each A-bounded subset sum problem SS(A) an 0(n) randomized greedy algorithm finds the correct maximum with probability 1/2 for almost all problem instances. Hence the A-bounded subset sum problem is almost everywhere randomly decidable in 0(n).