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I do not know how to solve this integer programming problem. $$\min_{w_{i,j}} \sum_{i=1}^{N}\sum_{j=1,j \neq i}^{N} w_{i,j}$$ $$s.t. \sum_{j=1,j \neq i}^{N} w_{i,j} \geq L, \forall i \in \left[ N \right]$$ $$w_{i,j} = w_{j,i}, \forall i \in \left[ N \right], j \in \left[ N \right], i \neq j$$ $$w_{i,j} \in \left[ S \right], \forall i \in \left[ N \right], j \in \left[ N \right], i \neq j$$

$L$ is a positive integer, $\left[ N \right]$ is a set of positive integers and $\left[ N \right] = \left\{ 1, 2, \dots, N \right\}$, $\left[ S \right]$ is a set of non-negative integers and $\left[ S \right] = \left\{0, 1, \dots, N \right\}$

Yuan Tan
  • 31

1 Answers1

1

You can use an integer programming package (e.g. mips in python) to solve the problem, first you need to define the variables in order to express the objective function and the constraints and define the parameter ($N, L$) values:

from sys import stdout
from mip import Model, xsum, INTEGER

N = 10
L = 2


model = Model()


w = [[model.add_var('w({},{})'.format(i, j), var_type=INTEGER, lb=0, ub=N)
      for j in range(N)] for i in range(N)]


model.objective = xsum(w[i][j]*(i!=j) for i in range(N) for j in range(N))


for i in range(N):
    model.add_constr(xsum(w[i][j]*(i!=j) for j in range(N)) >= L)


for i in range(N):
    for j in range(N):
        model.add_constr(w[i][j] == w[j][i])


model.optimize()


if model.num_solutions:
    stdout.write('\n')
    for i, v in enumerate(model.vars):
        stdout.write(str(int(v.x)))
        if i % N == N-1:
            stdout.write('\n')


solution


0000000020


0000002000


0000100100


0000010001


0010000100


0001000001


0200000000


0010100000


2000000000


0001010000

The above solution to be interpreted as $w_{1,9}=2$, $w_{2,7}=2$, $w_{3,5}=w_{3,8}=1$ etc.

Sandipan Dey
  • 2,111