Questions tagged [integer-programming]

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23
votes
3answers
1k views

What is known about solutions to sparse integer linear programming problems?

If I have a set of linear constraints in which each constraint has at most (say) 4 variables (all nonnegative and with {0,1} coefficients except for one variable that can have a -1 coefficient), what ...
21
votes
3answers
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 $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$...
14
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0answers
264 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 ...
13
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3answers
824 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 ...
12
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2answers
3k views

Integer programming with a fixed number of variables

The famous 1983 paper by H. Lenstra Integer Programming With A Fixed Number Of Variables states that integer programs with a fixed number of variables are solvable in time polynomial in the length of ...
11
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4answers
665 views

Is 0-1 programming with constant number of constraints polynomially solvable?

It was shown in the paper "Integer Programming with a Fixed Number of Variables" that integer programmings with constant number of constraints (or variables) are polynomially solvable. Does this hold ...
10
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2answers
576 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 ...
9
votes
2answers
373 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 ...
8
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2answers
915 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 ...
7
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1answer
111 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 ...
7
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0answers
272 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
121 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
150 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
votes
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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2answers
361 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 $\leq1$ ...
5
votes
0answers
57 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
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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
votes
1answer
403 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 ...
4
votes
2answers
181 views

On facets of 01-polytope

$0,1$-polytopse are fundamental objects in combinatorial geometry and comvex optimization. I am interested in the size of binary representation of hyperplanes to use in the framework of computational ...
4
votes
1answer
139 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 ...
4
votes
1answer
176 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?
4
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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 ...
4
votes
1answer
432 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 ...
4
votes
1answer
373 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 ...
4
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0answers
81 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
184 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
votes
0answers
248 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
702 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
180 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
votes
1answer
130 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
votes
1answer
186 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 ...
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 ...
3
votes
1answer
226 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}^...
3
votes
3answers
378 views

Facility location problem with a cost function

I'm struggling with a facility location problem. In its original form the problem is quite straightforward: Given a matrix of distances between cities, I have to pick a minimal number of centers from ...
3
votes
1answer
406 views

Applications and benchmarks for binary quadratic program algorithms

I have an algorithm that on all examples I was running finds an arbitrary approximation of global minimum of binary quadratic program. The algorithm find the minimum in polynomial time. Binary ...
3
votes
1answer
111 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 ...
3
votes
1answer
224 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 ...
3
votes
1answer
136 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 ...
3
votes
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 ...
3
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0answers
243 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_{...
2
votes
1answer
288 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 ...
2
votes
1answer
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
1answer
140 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 ...
2
votes
1answer
77 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 - ...
2
votes
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 ...
2
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0answers
60 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
387 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
140 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
96 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 ...