Questions tagged [integer-programming]

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14
votes
2answers
3k views

Integer programming with a fixed number of variables

The famous 1983 paper by H. Lenstra Integer Programming With A Fixed Number Of Variables states that integer programs with a fixed number of variables are solvable in time polynomial in the length of ...
4
votes
1answer
227 views

Is this edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{v\in V} ~d_{out}(v)\...
13
votes
3answers
887 views

Which Integer Linear Programs are easy?

While trying to solve a problem, I ended up expressing part of it as the following integer linear program. Here $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$ are all positive ...
5
votes
2answers
343 views

Is that edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{u\in V} ~\left( d_{...
22
votes
3answers
2k views

How fast can we solve a totally unimodular integer linear program?

(This is a follow-up to this question and its answer.) I have the following totally unimodular (TU) integer linear program (ILP). Here $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$ ...
5
votes
1answer
1k views

Reduction from SAT to 0,1 integer linear program with zero or one solutions

Probably this is well known. There is probabilistic reduction from SAT to Unique SAT (0 or 1 solutions). According to answer and comments derandomizing the reduction would imply $PH \subseteq \oplus ...
4
votes
1answer
473 views

Minimum Union-Sum Cost Path

I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
4
votes
1answer
412 views

FPT algorithm for mixed integer program

It is known that every integer linear program parameterized by the number of variables is FPT (fixed parameter tractable). Is every mixed integer program parameterized by the number of integer ...
2
votes
1answer
112 views

Do there exist two equivalent objective functions one of which can be approximated but another one cannot?

I have two equivalent problems A and B, meaning that the optimal solution of one must be the optimal solution of another one. However, it seems that problem A can be approximated but B cannot. Below ...
10
votes
2answers
600 views

Exact exponential-time algorithms for 0-1 programming

Are there known algorithms for the following problem that beat the naive algorithm? Input: A system $Ax \le b$ of $m$ linear inequalities. Output: A feasible solution $x^*\in \{0,1 \}^n$ if one ...
3
votes
3answers
438 views

Facility location problem with a cost function

I'm struggling with a facility location problem. In its original form the problem is quite straightforward: Given a matrix of distances between cities, I have to pick a minimal number of centers from ...