math - Algorithm for finding if set of numbers can add up to X, with constrained histogram -

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(motivation: consider problem have select sports team population of available players. each player has skill level proportional salary expectation, , want total of skill/salary level match total salary cap.)

i need write following function:

 bool possibleassignment(int n, int m, int t, vector<int> h);

the input constraints are:

  • 0 < n <= 50
  • 0 < m <= 50
  • 0 < t <= 2500
  • h.size() == n + 1
  • forall i, 0 <= h[i] <= m

possibleassign returns true iff array x of m ints can assigned following 3 constraints:

  1. forall i, 0 <= x[i] <= n
  2. forall v, number of elements of x value v <= h[v]
  3. the sum of x t

by algorithm or method can implement possibleassign?

this problem seems reduceable subset sum problem, or better known variant: knapsack problem, np-hard, there no known polynomial solution it.

however, seems t small enough, , luckily, there pseudo polynomial solution problem using dp.

because problem similar knapsack problem, i'd try reduce problem fit knapsack problem, , invoke dp algorithm finding best solution knapsack problem:

i'd first filter list, keep h[v] elements value v. now, set elemets follows:

value(x) = 1 weight(x) = x bag size = t

this going - giving maximal number of elements can assigned salary constraint t


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