Questions tagged [integer-programming]
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What is the complexity of Parametric Mixed Integer Linear Programming?
We know
$$\forall\bf y\in\mathbb Z^n:K\bf y\leq b$$
$$\exists\bf x\in\mathbb Z^m:A\bf x + B\bf y\leq c$$
is in $\bf P$ if $n,m$ are fixed from Kannan's result (refer page $1$ in reference).
What is ...
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19 views
Reducing Mixed Integer Quadratic Programming with only integer-real quadratic terms to Mixed Integer Linear Programming?
In Lenstra-Khachiyan's algorithm we solve for
$$(x,y)^T\in\mathbb Z^n\times\mathbb R^m$$
$$A(x,y)^T\leq b$$
by running Lenstra of the integer part and Khachiyan on the real part.
Suppose our program ...
1
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0answers
9 views
Size of solutions in integer programming
Given a linear integer program $Ax\leq b$ with $A\in\mathbb Z^{m\times n}$ and $b\in\mathbb Z^m$ known is there a polynomial time algorithm to give tight upper bounds for $\log_2\|x\|_\infty$ and $\...
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0answers
75 views
Fixed dimension Linear Integer Programming in $NC$
We know if fixed dimension linear integer programming is in $NC$ then integer $GCD$ is in $NC$. Is this the only non-trivial implication of fixed dimension linear integer programming in $NC$?
4
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1answer
388 views
Is there a counterexample to this work?
Is there a counterexample to this claim https://arxiv.org/abs/1610.00353? They claim a $O(n^6)$ LP model with simulations to support. I think asking validity is not a reasonable problem. However ...
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0answers
124 views
Convex mixed linear integer programming with real nuclear norm objective and linear integer objective
Khachiyan and Porkolab in 'Integer optimization on convex semialgebraic sets' gave an $O(ld^{ O(k^4)})$ algorithm to minimize a degree $d$ form with integer coefficients of binary length at most $l$ ...
3
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1answer
104 views
Integer programming: enforce the constraint that a subgraph contains at most $k$ connected components?
I'm considering integer programming on an variation of Steiner Forest Problem:
Given a graph $G=(V,E)$, a cost function: $c:E \rightarrow R^{+}$, a terminal set $T \subseteq V$, and a positive ...
4
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1answer
402 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 ...
5
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1answer
136 views
Anti bin packing
I have a (practically) unlimited amount of McDonald's coupons that I can use only if I shop for at least 1 money.
Thus I want to partition my family's meal into as many parts as possible that all ...
1
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0answers
87 views
Fixed dimension Integer programming minus LLL in fixed parameter $NC$?
If you remove LLL part then is remaining part of
a. Lenstra algorithm
b. Barvinok algorithm
in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$...
1
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0answers
105 views
Solving 0/1 integer programming and solving ACC-of-SYM circuits
I am referring to the proof of Theorem 1.4 in this STOC 2014 paper, https://arxiv.org/abs/1401.2444. In particular my question is about the argument that begins in the 8th line of page 9 where the ...
2
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0answers
57 views
Has Khachiyan/Porkolob's convex integer optimization been implemented?
Khachiyan and Porkolab in 'Integer optimization on convex semialgebraic sets' gave an $O(ld^{ O(k^4)})$ algorithm to minimize a degree $d$ form with integer coefficients of binary length at most $l$ ...
0
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1answer
40 views
How do minimally violated k-mod cuts work (intuitive explanation)?
As a background, I am not a specialist in theoretical computer science. But I have to take an exam with research-level optimization topics, and I have to learn it on my own, without lectures or tutors....
5
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1answer
174 views
Have fixed parameter integer program algorithms ever been implemented for research use?
Have any fixed parameter integer programming algorithms described in Integer programming with a fixed number of variables been implemented? Is there a reference code that researchers can use?
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0answers
180 views
What exactly did Lenstra prove on mixed integer linear program?
I studied Lenstra's paper https://www.jstor.org/stable/3689168. I have no clue what complexity he provides on Mixed Integer Programming (it is too terse and it is not a stand alone paper as he assumes ...
1
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1answer
193 views
Max weight travel on a graph with deadline
Given a deadline $D>0$ and a complete graph $K_n$ (with loops) in which each edge $e_{ij}$ has a weight $w(e_{ij}) \ge 0$ and a travel time $l(e_{ij}) > 0$. Starting from one of the nodes, we ...
1
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1answer
172 views
Fixed parameter tractable Integer Programming and $FPP$
Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $...
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0answers
58 views
Polynomial cases of 0 1 quadratic programm with linear constraints
A pseudo boolean function f:{0,1}^n-> R is defined as f(x)= x^tQx +cx where
Q is a symmetric matrix with null elements in the diagonal. Finding the minimum of this function is solvable in polynomial ...
3
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0answers
103 views
Fixed parameter Integer Programming circuit depth complexity
It is well known Lenstra's and Kannan's algorithm achieves $n$-variable parameter $L$-bit integer programming solvability in $O(n^nL)$ time and $O(L)$ space.
If implemented as an arithmetic circuit ...
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0answers
257 views
Under what conditions does an Integer Programming problem run in polynomial time?
Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
4
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1answer
129 views
Consequences of faster parameterized integer programming
Integer programming in $k$ variables can be done in $k^{O(k)}$ time and $O(k^c)$ space.
Is there any consequence if it can be done in $k^{O(k^\alpha)}$ time and $O(k^c)$ space for some $\alpha\in(0,1)...
3
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1answer
214 views
Space complexity of integer programming
Given $A\in\Bbb Z^{n\times k}$, $v\in\Bbb Z^n$ and variables $x_1,\dots,x_k$ given as a vector $x$ we know that solving $x\in\Bbb Z^k$ in $Ax\leq v$ is fixed parameter tractable. There is a ...
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2answers
348 views
On integer programming
Integer programming is NP-hard.
What is the status of integer programming problem that decides between existence of $\leq1$ solution and $>1$ solutions (note $0$ solutions falls in $\leq1$ ...
4
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1answer
132 views
Min cost set of edges to connect 2 subgraphs s.t dist of nodes between subgraphs <= K
I find myself with another graph problem that I can't find the name of. I was wondering if anyone was able to identify if this problem and any efficient algorithms to solve it are known.
The ...
2
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1answer
279 views
What are some example problems for integer programming that are *not binary*
I'm interested in NP-hard problems that have a "nice" integer-programming formulation (quadratic or linear, with quadratic or linear constraints) that is not binary.
Of course it is always possible ...
4
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0answers
230 views
Are there integer programs with small coefficients that only have large solutions?
It is well-known that if an integer linear program has a feasible solution, then it has a feasible solution whose bit size is polynomially bounded. For example, here is Theorem 13.4 from Papadimitriou ...
1
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1answer
267 views
Is finding an optimal solution to this Knapsack-like problem NP-hard?
Suppose our inputs are a set of objects with weights $w_1,...,w_n$. We have two separate sets of profits: $p_1,...,p_n$ and $v_1,...,v_n$. We wish to maximize $ \sum_{i=1}^{n} p_i(1-x_i)+\alpha_i ...
3
votes
1answer
224 views
Which is the approximation class of Minimum 0-1 integer programming if only non-negative integers are allowed?
I'm wondering which is the approximation class of Minimum 0-1 Integer Programming if only non-negative integers are used. That is:
minimize $c^T \cdot x$,
with $x\in \{0,1\}^n$ and $c\in (\mathbb{Z}^...
2
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0answers
374 views
How to prove integrality of LP with not totally unimodular matrix
I have a linear program (LP) for which the constraint matrix is NOT totally unimodular (TU). However, even though constraint matrix is small (14x20), extensive generation of random coefficients for ...
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2answers
400 views
NP completeness of linear $0-1$ assignment problem
Supposing we have a linear equation in $n^2$ variables with integer (negatives allowed) coefficients of at most $m$ bits each.
Partition $\Pi_1$ the variables into $n$ disjoint sets of $n$ variables ...
1
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1answer
197 views
On a Linearization of the Quadratic Assignment Problem
The Quadratic Assignment Problem formulated as an integer program:
\begin{align}
\mbox{minimize}\quad & \sum_{i=1}^n\sum_{j=1}^nc_{ij} x_{ij} + \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n ...
1
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0answers
130 views
Runtime of Gomory's Cutting Plane Algorithm
I read in several sources that the use of Gomory's cuts exclusively in Integer Programming was shown to be inefficient in practice when Gomory had created them. But later down the line they were shown ...
2
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1answer
76 views
Complexity of generating a pseudo-Boolean function
A pseudo-Boolean function is a mapping from $\mathcal{B}^n = \{0, 1\}^n$ to
$\mathbb{R}$.
Following is a pseudo-Boolean function.
$$s_1 s_4 - s_2 s_3 - s_3 s_5 - s_2 s_5 + s_1 + s_4 - s_1 s_3 - ...
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0answers
69 views
Is there a name for this unimodilarity-related property?
Consider an arbitrary integer linear program of the form
$\min f(x,y) \\
@ \ Ax + By \leq c\\
x,y \in \mathbb{Z}_+$
If you continuous-relax the integer constraint and still always get integer ...
1
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0answers
893 views
Is there a reduction from a 0-1 knapsack problem to the unbounded problem?
As we know, an unbounded knapsack problem could be described as:
$\max \sum_{i=1}^nc_1x_i$
s.t. $\sum_{i=1}^na_ix_i\le b$
$x_i\ge0,x_i\in\mathbb Z,i=1,\cdots,n$
And for an 0-1 knapsack problem, we ...
-1
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1answer
243 views
convertion into integer linear program for Ising spin state problem [closed]
I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem)
$$
maximise: \...
7
votes
1answer
108 views
Quanitifier Free Presburger Arithmetic: Upper bound on solution size?
DISCLAIMER: I had originally posted this to CS.SE, but I've deleted it and moved it here, since it received little attention, and I think it is a research level question.
According to this paper, if ...
6
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0answers
149 views
Are highly symmetric inequalities solvable over integers?
Suppose I have $n$ variables $x_1,\ldots,x_n$ that satisfy some inequalities that are highly symmetric, e.g., for all $S\subset [n], |S|=k$ we have $\sum_{i\in S} f(x_i,k)\le \sum_{i\in [n]} g(x_i,k)$,...
4
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1answer
355 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 ...
5
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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 ...
5
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0answers
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Interpolating the Tutte polynomial at the values of two hyperbolas
In a MO question
basically I asked when the Tutte polynomial of planar graph
can be uniquely determined by the polynomially computable values at the special points
and at the two hyperbolas. The ...
2
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0answers
94 views
Linear ordering of bipartite tournament graphs
I have an ordered bipartite graph such that every node of the first node set is connected to every node in the second node set by an edge of a given direction (i.e., a so-called bipartite tournament ...
3
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1answer
184 views
What is known about this binary representation polytope?
Consider the following set.
$S_n =\{ (x,y) ~|~x \in \mathbb{Z}_+~\wedge~ y \in \{0,1\}^n ~\wedge~x=\sum_{i=0}^{n-1} 2^i y_i \}$
$S_n$ is a collection of pairs $(x,y)$, where $x$ is an integer ...
1
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1answer
76 views
Small number of inequalities defining integer hull of integer programming problem
In integer programming problem, we often want to relax the integer programming problem to linear programming problem. So we want to find the integer hull of the problem. The number of inequalities to ...
2
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1answer
79 views
Paper about the upper bound of the number of inequalities to describe the Integer hull of a polyhedron
I am interested in the upper bound of the number of inequalities to describe the integer hull of a polyhedron. That is, given an integer programming problem with n inequalities which construct a ...
4
votes
2answers
256 views
Lattice-based algorithms in practice
Are there any applications of lattice-based algorithms other than those in cryptography and integer programming?
Could someone state a few papers where
the primary algorithms use lattice-based LLL ...
2
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0answers
217 views
Quadratic Binary Optimization formulation of Steiner Tree problem
can someone point out to me a solution or give advice on how to formulate as efficiently as possible in terms of number of bits the minimum Steiner tree problem as a 0-1 quadratic optimization problem?...
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2answers
372 views
Exact exponential-time algorithms for 0-1 programs with nonnegative data
Are there known algorithms for the following problem that beat the naive algorithm?
Input: matrix $A$ and vectors $b,c$, where all entries of $A,b,c$ are nonnegative integers.
Output: an ...
10
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2answers
575 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 ...
3
votes
1answer
190 views
Efficient flow problem for a complex integer program
I have a bunch of marbles each with some weight (they can also have negative weights). I want to pick the nodes such that the weight is maximized. The only rule is that if I pick X1 and X2 I have to ...
