Questions tagged [computational-complexity]

For questions about the theoretical runtime needed for solving computational problems, often measured in the size of the input. This includes questions about whether polynomial time algorithms exist, NP-hardness, among others.

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3
votes
1answer
67 views

confusing results of two models with different complexity

i have two models that address the same problem. the first one is : the second one is: for different instances for the same size (n=30) i found the following results ( the first column on the left ...
5
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1answer
204 views

How to determine the size of a model?

I want to know about the number of variables and constraints of this formulation (exp: $o(n)$ variables and constraints or $o(n^2)$ ....). Is the number of variables $\mathcal O(n^3)$ because we have ...
6
votes
3answers
119 views

How to find all descendant vertices of all vertices in a big DAG (Directed acyclic graph)?

A simple algorithm may be traverse all vertices, and perform DFS for every vertex. However, the computational complexity is $O(n(n+m))$, where $n$ and $m$ are the number of vertices and edges in the ...
4
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1answer
73 views

Max flow problem with piece-wise costs

This question is a variant of a question I posted earlier and also fixes some typos in the earlier post (Complexity \ Reference request for variant of max flow problem). Some of the changes are ...
6
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1answer
57 views

Complexity \ Reference request for variant of max flow problem

In the standard max cost flow problem with arc capacities, the cost of using an arc is proportional to the flow through the arc. For example, if $f_{uv}$ is the flow through the arc $(u,v)$, then the ...
4
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0answers
44 views

Complexity of solving a certain commodity flow problem

Does anyone know the complexity of obtaining the optimal solution to the integral multi-commodity network flow problem with unit demands, integral capacities, but the cost of using an arc varies ...
5
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2answers
145 views

Polynomial algorithm for a special ILP problem

Given the following problem: \begin{align} & z=\min \sum_{ij} x_{ij}\\ \text{s.t.} & \quad \sum_i d_{ij} x_{ij} \ge s_j, \quad \forall j \tag1 \\ & \quad \sum_j x_{ij} \le 1, \quad \...
3
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2answers
569 views

Fast algorithm for Transportation Problem in Python?

Transportation Problem can be solved with simplex algorithm, but it's time demanding. I'm wondering if there exist a specific Python-implemented algorithm with low complexity. Thanks
5
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1answer
315 views

Polynomially solvable cases of zero-one programming

I am dealing with a problem having two types of variables: binary variables, and continuous variables. In some cases, the continuous variables are not used, and so the problem contains those binary ...
5
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0answers
49 views

Complexity of determining whether a LP or MIP is infeasible

What is the best complexity for the worst case scenario and the algorithm associated with it to determine if a linear programming (LP) is infeasible ? Further, what if we consider a mixed integer ...
5
votes
1answer
56 views

If a problem is inapproximable for $(2-\epsilon)$, can we conclude there exists no PTAS for it?

If we prove that: The existance of a $(2-\epsilon)$-approximation algorithm for Problem P1 implies $P = NP$, can we conclude: There exists no PTAS for Problem P1, and so P1 is APX-hard?
7
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1answer
121 views

Can this algorithm be considered polynomial?

Let us assume that an optimization algorithm requires $\mathcal{O}(n^{\log1/\epsilon})$ flops to find a solution $\bar{X}$ such that $$\| \bar{X} - X^{\star}\| \leq \epsilon$$ where $\epsilon < 1$...
12
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1answer
130 views

Re-calculating shortest path in slightly altered graph

I was wondering if someone has come across this before and/or has a smart idea for the following: I have a directed graph $G$ with costs $c$ associated with the arcs, and I know the shortest path $P^...
8
votes
1answer
102 views

Complexity comparision between purely BLP and MILP problems?

Could someone please comment and answer on the complexity of purely binary linear programming (BLP) and mixed-integer linear programming (MILP)? In MILP, we have both binary and continuous variables ...
5
votes
1answer
161 views

Solving a variant of multiple knapsack problem/ generalized assignment problem

Consider $m$ knapsack and $n$ items. With each knapsack $j$ associated a capacity $c(j)$ and with each item $i$ associated a profit $p(i,j)$ (that depends on the knapsack, so it's not exactly the ...
11
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2answers
561 views

MILP: is it NP-complete or NP-hard?

The pieces of information I get online are sometimes confusing. Someone says MILP problems are NP-hard, and somewhere else I found the claim that MILP problems are NP-complete. Can someone please ...
8
votes
2answers
792 views

Complexity of LP and MILP Problems?

My original problem is an MILP. I make it an LP by relaxing the integer variables. Can someone please comment on the complexity, solvability and optimality of MILP and LP problems, in general? Is ...
11
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2answers
215 views

Generalized Assignment Problem as the sub-problem

I was wondering what is the state-of-the-art for solving the Generalized Assignment Problem (GAP) and if there are special cases that are polynomially solvable? Moreover, is there any usage of this ...
19
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2answers
802 views

How do we decide/plan an SLA for an NP-hard optimization process running in production?

How do you decide or plan an SLA (Service Level Agreement) for an application that depends on an optimization process when the problems you deal with are NP-hard? That is, if you are developing an ...
17
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3answers
1k views

Can an integer optimization problem be convex?

I'm trying to wrap my head around an apparent paradox that I've come across while trying to learn more about optimization algorithms: On one hand several sources state that convex optimization is ...
-4
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2answers
366 views

Are there NP hard problems solved in P time?

Does anyone know of a problem previously believed to be NP hard, to be solved nowadays in polynomial time optimally?
7
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1answer
607 views

Why is the Ellipsoid Method of polynomial complexity?

We know that the ellipsoid method is proven to be of polynomial complexity. However, as far as I can tell we may need to add exponentially many feasibility cuts in order to solve the LP (or prove no ...
14
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2answers
796 views

State-of-the-art algorithms for solving linear programs

Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances)....
16
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6answers
3k views

Does the problem of P vs NP come under the category of Operational Research?

I am enrolled in an Operational Research program. I am also interested in Algorithms, and I wish to know whether "P vs NP" is a common point in both of the fields, so that the effort put in ...
12
votes
1answer
522 views

Complexity of verifying optimality in (mixed) integer programming

I looked around for a while, but I couldn't find a precise answer to the following question. If I'm given a candidate solution for a (mixed) integer (convex) program, what's the complexity of ...
21
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1answer
284 views

Polynomially solvable problems with exponential extension complexity

The maximum matching problem is solvable in polynomial time using Edmonds' blossom algorithm. However, unlike for example the spanning tree polytope, it has been proven fairly recently that the ...
9
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0answers
84 views

Modelling resource dependency in the assignment problem

The assignment problem is well-studied and has a nice polynomial time algorithm. I'm interested in an extension of this problem where all edges are in a certain group and taking multiple edges from ...
11
votes
2answers
189 views

Is deciding the presence of mixed-integer points in the relative interior of a polyhedron in NP?

Given $P = \{x\in\mathbb R^n: Ax \leq b\}$, I want to decide if $(\mathbb Z^\ell \times \mathbb R^{n-\ell}) \cap \operatorname{relint}(P)$ is non-empty. Is this problem in NP? One idea is to check ...
11
votes
2answers
218 views

Can SOCPs approximate better than LPs?

Are there any classes of NP-hard combinatorial optimization problems where Second order cone programs (SOCP) gives a better approximation than linear programs (LP)? I am looking for results in the ...
5
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0answers
97 views

Maximum eigenvalue across subsamples

I have an $N$-dimensional vector of data, say $X_{t}$, with $1 \leq t \leq T$. Of this vector $X_{t}$, I want to consider sub-vectors, say $X_{t}^{b}$, which are $m$-dimensional combinations of ...
12
votes
1answer
277 views

When is the original BFGS algorithm still better than the Limited-Memory version?

I have been going through Andrew NG's original data science course on Coursera. I learned the BFGS algorithm at some point in my OR education, but not the Limited Memory version that Andrew NG focuses ...
13
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3answers
2k views

Are there any efficient algorithms to solve the longest path problem in networks with cycles?

I have a directed social network and as a preprocessing step I need to calculate the longest path lengths for each node. Longest path problem is NP-hard as far as I know but I've seen dynamic ...
11
votes
1answer
83 views

Computational complexity to compute an IIS

How hard is it to compute an irreducible infeasible subset (IIS) for a linear program? What about an integer program (e.g., removing the integrality constraint on a single variable may be enough to ...