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Integer Program where an integer optimal solution is an extreme point of the LP Relaxation

Let RLP $\equiv$ (Linear relaxation of the Integer Program). In general, Integer Programming is NP-hard. $~$And often, an integer optimal solution will lie in the interior of the feasible space of the ...
Shuxue Jiaoshou's user avatar
1 vote
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What is a "strongly complementary pair" of primal/dual solutions to a linear program?

While trying to understand this paper by Hammer, Hansen and Simeone, I came across some terminology I was unfamiliar with: the notion of a "strongly complementary pair". For a linear program ...
RubenVerhaegh's user avatar
2 votes
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69 views

Parallel complexity of fixed dimension fixed constraints integer programming

Papadimitriou in https://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf shows ILP is fixed parameter tractable in number of constraints and Lenstra in https://people.csail.mit....
Turbo's user avatar
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Separation oracle for breaking cycles in directed graph

I am working on a directed graph problem and am collaterally interested to know whether there is a separation oracle for the following set of linear constraints. We are given a directed graph $G$ ...
reservoir's user avatar
2 votes
0 answers
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Does the standard 4/3 integrality gap for TSP example work for Euclidean TSP?

Given a graph $G=(V,E)$, costs $c \in \mathbb{R}^E$ the TSP problem is to compute a min cost tour of the graph. The LP is min $ c^tx $ $x(\delta(S)) \geq 2 \ \ \ \ \forall S \subset V $ $x(\delta(v)...
Hao S's user avatar
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1 answer
136 views

Hardness of computing the dimension of an integral polytope?

Given a set of linear inequalities $Ax \leq b$ let $P = \text{conv}\{x \in \{0,1\}^n \mid A x \leq b \}$ be the convex hull of all binary vectors that satisfy the given inequalities. I am interested ...
badboul's user avatar
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1 answer
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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 ...
user avatar
3 votes
1 answer
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How hard is this combinatorial optimisation problem?

Suppose we have multiple intervals $R_1,R_2,...,R_i$ of non-negative integers. These intervals may overlap and we use $R_h(\mathrm{median})$ to denote the median integer in the $h$-th interval $R_h$, ...
user avatar
2 votes
1 answer
209 views

Characterization of integral polyhedra

A rational polyhedron $P \subseteq \mathbb{R}^n$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $P = conv(P \cap \mathbb{Z}^n)$. Equivalently, $P$ is integral if ...
Karagounis Z's user avatar
8 votes
3 answers
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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_{...
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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)\...
maxdan94's user avatar
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(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 ...
GBathie's user avatar
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1 answer
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Ensuring integral maximizer from integral linear program

An integral linear program is one that has a maximizer that is integral. Sometimes it's possible to prove that a particular LP has this property, for example by proving that it's constraint matrix is ...
orlp's user avatar
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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, ...
Karagounis Z's user avatar
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158 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 ...
Karagounis Z's user avatar
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151 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 ...
Mengfan Ma's user avatar
1 vote
1 answer
91 views

Name of (and solution to) this generalization of linear assignment

I would like to know if the following problem is known and has any efficient solution. Given an $n\times n$ score matrix $S$. Find the best $a$ elements, in terms of their sum of scores, such that no ...
f10w's user avatar
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2 votes
0 answers
98 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 ...
Xiao's user avatar
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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, $\...
Alex Coventry's user avatar
5 votes
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87 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,...
AmeerJ's user avatar
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1 vote
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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 $\...
Turbo's user avatar
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4 votes
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Problems which will be in $NC$ if 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 $NC$?
Turbo's user avatar
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1 answer
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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 ...
Turbo's user avatar
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0 answers
138 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 $l$ ...
Turbo's user avatar
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4 votes
1 answer
242 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 ...
Mengfan Ma's user avatar
4 votes
1 answer
526 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 ...
mnmp's user avatar
  • 175
6 votes
1 answer
204 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 ...
domotorp's user avatar
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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, $m$ ...
Turbo's user avatar
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1 vote
0 answers
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 ...
gradstudent's user avatar
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2 votes
0 answers
69 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 $l$ ...
Turbo's user avatar
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0 votes
1 answer
90 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....
rumtscho's user avatar
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4 votes
1 answer
196 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?
Turbo's user avatar
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5 votes
0 answers
273 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 ...
Turbo's user avatar
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1 vote
1 answer
316 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 ...
Helium's user avatar
  • 463
1 vote
1 answer
244 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 $...
Turbo's user avatar
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1 vote
0 answers
64 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 ...
Arty's user avatar
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3 votes
0 answers
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 ...
Turbo's user avatar
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7 votes
0 answers
354 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 ...
Turbo's user avatar
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3 votes
1 answer
138 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)...
Turbo's user avatar
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4 votes
1 answer
387 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 ...
Turbo's user avatar
  • 12.6k
5 votes
2 answers
407 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 $\leq1$ ...
Turbo's user avatar
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3 votes
1 answer
249 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 ...
Erotemic's user avatar
  • 133
3 votes
1 answer
527 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 ...
Lagerbaer's user avatar
  • 319
5 votes
0 answers
374 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 ...
Dominik Peters's user avatar
1 vote
1 answer
320 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 ...
mich's user avatar
  • 389
3 votes
1 answer
252 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}^...
EXPTIME-complete's user avatar
2 votes
0 answers
469 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 ...
maksay's user avatar
  • 31
0 votes
2 answers
653 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 ...
Turbo's user avatar
  • 12.6k
1 vote
1 answer
264 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 ...
f10w's user avatar
  • 241
2 votes
0 answers
197 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 ...
Sidharth Ghoshal's user avatar