Maximum Box Problem on Stochastic Points

Abstract Given a finite set of weighted points in $${\mathbb {R}}^d$$ R d (where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that sum weights it contains is maximized. We consider each point input has probability being present final random set, and these events are mutually independent; then, total weight variable. aim to compute both this variable at least given parameter, its expectation. show even $$d=1$$ = 1 computations #P-hard, give pseudo-polynomial time algorithms case where integers bounded interval. For $$d=2$$ 2 , we colored red or blue, have $$+1$$ + blue $$-\infty $$ - ∞ . The number covered with not containing any point. prove above two also polynomial-time algorithm computing exactly points, no point, plane.