TITLE: Combinatorial Problem
QUESTION [7 upvotes]: Any ideas or references on how to approach this problem? Every element in a set has a parameter $p_i \geq 0$ and $c_i \geq 0$. The objective is to find a subset which maximizes
$\prod_{i \in S} p_i + \sum_{i \in S} c_i$
I tried to prove complexity results but was unsuccessful. My guess is that it is NP-Hard, but I cannot find a reduction from any known problem. 
A greedy heuristic which sequentially adds elements which give the highest increase works fine, but can give arbitrarily bad results.

REPLY [14 votes]: It is actually an easy one. Suppose that you were allowed to take any part of each item adding $tc_i$ and multiplying by $p_i^t$ for any $t\in[0,1]$. First, you should take everything that has $p_i\ge 1$ (because it never hurts). Then you have to maximize $C\prod_j e^{-b_j t_j}+\sum_j t_j c_j$ where $b_j=-\log p_j>0$. Now the points $(\sum t_j b_j, \sum t_j c_j)$ form a convex polygon and you know its boundary (half of it comes from arranging $c_j/b_j$ in the decreasing order and the other half from arranging them in the increasing order). Moreover, the functional $(x,y)\mapsto Ce^{-x}+y$ is convex, so its maximum is attained at some vertex on the boundary (which correspond to some subsets) and you are done in $n\log n$ time most of which is spent on sorting. 
Now, if the task were to minimize that quantity, that would be worse than the knapsack packing problem.