partition-problem

I've came across some similar problems to this one in the past, and I still haven't got good idea how to solve this problem. Problem goes like this: You are given an positive integer array with size n <= 1000 and k <= n which is the number of contiguous subarrays that you will have to split your array into. You have to output minimum m, where m = max{s[1],..., s[k]}, and s[i] is the sum of the i-th subarray. All integers in the array are between 1 and 100. Example : Input: Output: 5 3 >> n = 5 k = 3 3 2 1 1 2 3 Splitting array into 2+1 | 1+2 | 3 will minimize the m. My brute force idea was to

Given a set A containing n positive integers, how can I find the smallest integer >= 0 that can be obtained using all the elements in the set. Each element can be can be either added or subtracted to the total. Few examples to make this clear. A = [ 2, 1, 3] Result = 0 (2 + 1 - 3) A = [1, 2, 0] Result = 1 (-1 + 2 + 0) A = [1, 2, 1, 7, 6] Result = 1 (1 + 2 - 1 - 7 + 6) You can solve it by using Boolean Integer Programming. There are several algorithms (e.g. Gomory or branch and bound) and free libraries (e.g. LP-Solve) available. Calculate the sum of the list and call it s. Double the numbers

Partition problem is known to be NP-hard. Depending on the particular instance of the problem we can try dynamic programming or some heuristics like differencing (also known as Karmarkar-Karp algorithm). The latter seems to be very useful for the instances with big numbers (what makes dynamic programming intractable), however not always perfect. What is an efficient way to find a better solution (random, tabu search, other approximations)? PS: The question has some story behind it. There is a challenge Johnny Goes Shopping available at SPOJ since July 2004. Till now, the challenge has been