Questions tagged [integer-programming]

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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 ...
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 $\ell,m,n_{1},n_{2},\ldots,n_{\ell},c_{1},c_{2},\ldots,c_{m},w$...
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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 ...
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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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 ...
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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 $\leq1$ ...
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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 ...
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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 $... 1answer 403 views Is there a counterexample to this work? Is there a counterexample to this claim https://arxiv.org/abs/1610.00353? They claim a$O(n^6)$LP model with simulations to support. I think asking validity is not a reasonable problem. However ... 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 ... 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 ... 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? 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 ... 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 ... 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 ... 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$NC$? 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 ... 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 ... 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 ... 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 ... 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)...
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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 ...
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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 ...
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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 ...
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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 ...