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0
votes
1answer
41 views

How to reformulate my problem as a mixed-integer quadratic problem

I have an unknown $n$-dimensional vector $x$ whose analytical expression depends on the following sum $x = z + Ba$ where the vector $z$ and the matrix $B\in \mathbb{R}^{n\times s}$ are given. So the ...
0
votes
0answers
60 views

Is this problem a knapsack problem?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: $\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ ...
1
vote
0answers
83 views

Is this NP-Hard problem? [closed]

I have the following problem. maximize $\sum\limits_{k=1}^Lx_k$ subject to: $\mathbf{x}^T\mathbf{A}~ \tilde{\mathbf{x}_i} \geq 0,~~ \forall~ i\in\{1, 2, \cdots, L\}.$ where, $~\mathbf{x}^T = (x_1, ...
5
votes
2answers
204 views

Find a binary matrix so that no vector from {-1,0,1}^n is in its kernel

Given integers $n,m$, I want to find a $m \times n$ binary matrix $X$ such that there does not exist any non-zero vector $y \in \{-1,0,1\}^n$ with $Xy=0$ (all operations performed over $\mathbb{Z}$). ...
8
votes
1answer
251 views

Poly-time reduction from ILP to SAT?

So, as is known, ILP's 0-1 decision problem is NP-complete. Showing it's in NP is easy, and the original reduction was from SAT; since then, many other NP-Complete problems have been shown to have ILP ...
10
votes
0answers
527 views
+100

Is it NP-hard to fill up bins with minimum moves?

There are $n$ bins and $m$ type of balls. The $i$th bin has labels $a_{i,j}$ for $1\leq j\leq m$, it is the expected number of balls of type $j$. You start with $b_j$ balls of type $j$. Each ball of ...
2
votes
3answers
1k views

Are all Integer Linear Programming problems NP-Hard?

As I understand, the assignment problem is in P as the Hungarian algorithm can solve it in polynomial time - O(n3). I also understand that the assignment problem is an integer linear programming ...
2
votes
0answers
28 views

Why are optimization problems always NP-hard and not NP-complete and what does this mean for other levels of the polynomial time hierarchy? [duplicate]

I have read that optimization problems cannot be $\mathcal{NP}$-complete, but are always classified as $\mathcal{NP}$-hard. When a problem is NP-complete, I know it is contained in $\mathcal{NP}$P. ...
8
votes
1answer
88 views

Hardness of Approximating 0-1 Integer Programs

Given a $0,1$ (binary) integer program of the form: $$ \begin{array}{lll} \text{min} & f(x) & \\ \text{s.t.} &A\vec{x} = \vec{b} & \quad \forall i\\ &x_i\ge 0 & \quad \forall ...