Questions tagged [integer-programming]

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NCness of unique solution fixed dimension integer programming

Consider the promise problem of fixed dimension integer programming with promise being that there is exactly one witness. The problem is in $TFNP$. If the promise problem is in $FNC$ then integer $GCD$...
4 votes
0 answers
118 views

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 $NC$?
-1 votes
0 answers
70 views

Theorems about integrality gap for set cover linear programs

What theorems about the integrality gaps of linear problems of the form $$ \begin{array}{c} \textbf{min} \ \ c^t x \\ \sum_{ v \in C} x_v \geq 1 \ \ \forall C \in \mathcal{C} \\ x \geq 0 \end{...
4 votes
1 answer
380 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 ...
0 votes
0 answers
93 views

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 $G$ ...
2 votes
0 answers
71 views

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)...
1 vote
1 answer
116 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 ...
6 votes
1 answer
202 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 ...
2 votes
1 answer
130 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 ...
3 votes
1 answer
352 views

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 $R_h$, ...
2 votes
1 answer
176 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 ...
6 votes
2 answers
466 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_{...
6 votes
1 answer
297 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)\...
11 votes
2 answers
3k views

Integer Factoring via Lattice Reduction?

I found a paper titled "Factoring integers and computing discrete logarithms via diophantine approximation" by C. P. Schnorr from 1993. It looks like a probabilistic method with expected polynomial ...
3 votes
0 answers
134 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 ...
1 vote
1 answer
110 views

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 ...
1 vote
0 answers
114 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 vote
0 answers
153 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 ...
1 vote
1 answer
87 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 ...
1 vote
0 answers
134 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 ...
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 ...
2 votes
0 answers
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, $\...
5 votes
0 answers
84 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,...
1 vote
0 answers
12 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 $\...
4 votes
1 answer
501 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 ...
3 votes
1 answer
496 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 ...
5 votes
1 answer
438 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 ...
0 votes
0 answers
136 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 $l$ ...
4 votes
1 answer
230 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 ...
2 votes
1 answer
2k views

Solve a simple system of linear inequalities in natural numbers

I want to find a solution to a system of linear inequalities of the following form \begin{aligned} a_1 + b &\ge a_2 \\\ \vdots \\\ a_4 + c &\ge a_1 \end{aligned} where $a_i \in \mathbb N \...
2 votes
0 answers
65 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 $l$ ...
1 vote
0 answers
108 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 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 ...
0 votes
1 answer
79 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....
4 votes
1 answer
195 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?
13 votes
3 answers
923 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
0 answers
256 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 vote
1 answer
287 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 vote
1 answer
241 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 $...
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 ...
1 vote
0 answers
62 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 ...
7 votes
0 answers
338 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 ...
22 votes
3 answers
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 $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$ ...
3 votes
1 answer
137 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)...
5 votes
2 answers
398 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 $\leq1$ ...
2 votes
1 answer
144 views

Can Lenstra's algorithm output all feasible solutions in O^*(f(k)) time where k is the number of variables and f is a computable function in k?

It is well-known that Lenstra's famous algorithm (presented in the paper Integer programming with a fixed number of variables) can solve an ILP problem in $O^*(f(k))$ time where k is the number of ...
1 vote
1 answer
2k views

Difference between Multiple Knapsack problem and Multidimensional Knapsack Problem

What is the difference between Multiple Knapsack problem and Multidimensional Knapsack Problem? (http://en.wikipedia.org/wiki/List_of_knapsack_problems#Multiple_constraints) According to the ...
3 votes
1 answer
236 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 ...
5 votes
1 answer
477 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 ...
1 vote
1 answer
301 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 ...