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Questions tagged [integer-programming]

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14
votes
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269 views

Any example of an unsatisfiable integer program with non constant Rank Lower bounds for LS+ cuts but with short LS+ refutations?

Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear ...
7
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0answers
308 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 ...
7
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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 ...
7
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0answers
123 views

Deterministic Parallel Algorithm for ILP with small number of variables and small coefficients

Given a set of $n$ linear inequalities in $d$ variables where the coefficients are integers of size bounded by $O(\log{n})$ is there a known deterministic parallel algorithm that runs in time $(d\log{...
6
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0answers
151 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 $\sum_{i\in S} f(x_i,k)\le \sum_{i\in [n]} g(x_i,k)$,...
5
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0answers
74 views

Relaxed minimum dominating set

(I moved this question from cs exchange to here, because it might be more on the topic here) Let $G=(V,E)$ be a directed graph with $n$ nodes. For a subset of nodes $S\subseteq V$, let $\mathcal{N}(S,...
5
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0answers
59 views

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 ...
5
votes
0answers
103 views

Quantized Unbounded Flow

I am interested in the following flow problem, since it turns out to be equivalent to a more general problem. INPUT: A graph where each edge $e$ has an integer multiplier $q_e$, and a lower bound $...
4
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0answers
86 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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0answers
200 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 ...
4
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0answers
291 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 ...
4
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0answers
1k views

How to determine proper rounding in linear programming relaxations?

Recall that in the vertex cover problem we are given an undirected graph ${G=(V,E)}$ and we want to find a minimum-size set of vertices ${S}$ that “touches” all the edges of the graph, that is, such ...
4
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0answers
186 views

LP-type vs. Approximation

I'm interested in an computational geometry problem that's sensibly expressed as an infinite dimensional 0-1 integer program. I'm not worried about finding an actual minimum for the objective ...
3
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0answers
104 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 ...
3
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0answers
247 views

Slightly Faster Exponential Algorithm for Integer Programming with Multi-linear Variables

Integer programing is one of the most narutal optimization tools. As an analogy of DNF or CNF in the Boolean function theory, we can consider the following equation. $x_{1}x_{2}x_{3}+$ $x_{3}x_{4}x_{5}...
2
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0answers
92 views

(Integer) Linear Program formulation of planarity?

Q: Is there an efficient (I)LP formulation of planarity? More specifically, I am looking for a set of constraints that are satisfied by exactly all planar graphs on $n$ vertices, in order to optimize ...
2
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0answers
97 views

An integer programming on maximum number of triangles

Consider three distinct sets $A, B, C$ of $N$ elements respectively. For each pair of sets, say $A,B$, we introduce an variable for each combination of values $a,b$ as $x_{ab}$. Similarly, we ...
2
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0answers
30 views

Finding shortest calculation of the sum of a subset of a group, given sums for other previously summed subsets

Say $S=\{g\in G\}$ is a set of elements in an abelian group $G$ whose group operation $(+)$ is expensive to compute. Given a subset $T\subset S$, we want to compute the sum of $T$'s elements, $\...
2
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0answers
64 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 $l$ ...
2
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0answers
411 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 ...
2
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0answers
164 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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0answers
100 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 ...
2
votes
0answers
223 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?...
2
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0answers
81 views

General covering approximation

Consider the following integer program (general covering): $\min c \cdot x$ subject to $Ax \ge b$, where all entries in $A, b, c$ are nonnegative and $x$ is required to be nonnegative and integral. ...
1
vote
0answers
92 views

Is the edge cover polytope integral on graphs with self-loops?

It is well known that the edge cover polytope is integral on simple graphs. I am wondering whether this also holds for graphs with self-loops. Here is a Linear Relaxation of the edge cover polytope, ...
1
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0answers
84 views

Separation oracle for hitting all small cut on a graph?

We are given as input an undirected graph $G=(V,E)$, weights $w_e \ge 0$ for all $e\in E$ and an positive integer $k$. We aim to select a set of edges with the minimum weight, such that the cut set of ...
1
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0answers
11 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 $\...
1
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0answers
104 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 $n$ dimension, $m$ ...
1
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0answers
110 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 ...
1
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0answers
59 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 ...
1
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0answers
957 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 ...
0
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0answers
76 views

Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

Crossposted at MathOverflow A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$. $\mathbf{1}$ is ...
0
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0answers
29 views

Optimum partitioning of vertices into mutually disjoint subsets in a weighted graph

tl;dr I'm trying to partition my students into groups with respect to their preferences, i.e. they can declare if they want to be with someone in a group or if they do not want to be with someone in a ...
0
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0answers
47 views

what is the LP gap of this particular non metric facility location problem in planar graphs?

Suppose I have a facility location problem with all service costs 0 (or infinity (in particular service costs are not metric)) such that the edges $ij$ for which facility $i$ can service client $j$ ...
0
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0answers
114 views

Is the matching polytope integral?

In this document https://courses.engr.illinois.edu/cs598csc/sp2010/Lectures/Lecture9.pdf they prove the integrality of the matching polytope using the integrality of the perfect matching polytope. The ...
0
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0answers
134 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 $l$ ...