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.
80 questions
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Johnson's Rule and Integer Programming Formulations: What Special Matrix Structure Enables Its Efficiency?
Johnson's Rule efficiently solves the two-machine flow shop scheduling problem to minimize makespan with remarkable computational simplicity.
I suspect that this effectiveness might imply the ...
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Subset Sum Variant with Positional Weighting: Strong NP-Hardness or Pseudo-Polynomial Algorithm?
I'm studying a variant of the classical Subset Sum problem and would appreciate insights on its complexity.
As background:
The classical Subset Sum problem with positive integers is known to be ...
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Efficient Approaches for Separable Concave Quadratic Minimization over Linear Constraints?
I am a network engineer working on optimization problems for network resource allocation. Recently, I have encountered a specific optimization challenge that I would appreciate expert guidance on.
The ...
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Can a sparse large LP be solved on a laptop?
I have a Linear Program $Ax\leq b$ where $A\in\mathbb Z^{n\times m}$, $b\in\mathbb Z^m$ and $x\in\mathbb R^n$ with $5\times10^5$ to $2\times10^6$ variables and the matrix $A$ has roughly $10^7$ to $10^...
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Reference on why finding a feasible solution to the Set Partitioning problem is NP-Hard?
The set partitioning problem is an integer problem that looks as simple as this problem
$\begin{array}{*{20}{c}}
{\bf{P_2}\mathop {\min }\limits_{{x_i} \in \left\{ {0,1} \right\}} }&{{x_1} + {x_2} ...
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Does turning the $\ge$ to $=$ sign increase the hardness of solving the Set-Cover problem?
The set cover problem, a well-known combinatorial issue, involves selecting the fewest number of sets from a collection $ S = \{s_1, \dots, s_m\} $, covering a universe $ U = \bigcup_{i=1}^m s_i $. ...
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Computational complexity of set cover problem with a fixed number of decision variables?
This paper show that integer programs with a fixed number of variables are solvable in time polynomial in the length of the data.
"In this section we show that the integer linear programming ...
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How to correctly interpret the $\ln(n)$ approximation ratio of the set cover problem under its integer formulation context?
The wikipedia article of the set cover problem stated the following point regarding the inapproximability of the greedy method "When $n$ refers to the size of the universe.... it cannot be ...
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Optimal way to formulate a piecewise linear function
I am working on LP problem whose objective function includes a piecewise linear function. I would like to figure out the optimal way to formulate the piecewise linear function in order to minimize the ...
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My Professor couldn't complete the model for this optimization problem. how do i model this problem?
(Edit: there was a slight translation error, to be clear, we tried this since the start with Binary variables (IP), even then we couldn't crack it)
I'm an undergraduate in Industrial Engineering and ...
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Is 0-1 knapsack problem still NP-Hard (1) with an equality constraint and (2) when all the weights in the constraint are equal to one?
Is 0-1 knapsack problem still NP-Hard with an equality constraint?
$$\begin{aligned}
& \text { maximize } \sum_{i=1}^n v_i x_i \\
& \text { subject to } \sum_{i=1}^n w_i x_i = W \text { and } ...
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Complexity of cardinality constrained maximum weight independent set problem
Given a graph with a set of nodes and edges, the goal of the maximum independent set problem is to find the maximum number of vertices where no two vertices are adjacent. This is well-known NP-hard ...
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Interior-point computational complexity for SDP
I am trying to find the complexity of the semidefinite programming (SDP) problem for my specific instance, but I’m facing some problems. I found in the literature that the complexity of the SDP ...
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How many variables and constraints can modern mixed integer programming solvers handle?
I originally asked a question here and they suggested that I crosspost it to the OR stack exchange, so that is what I am doing (hopefully correctly?). Here is the question I asked there:
"I know ...
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Polynomial time separation and optimization for an LP with exponential columns and rows?
We know the fundamental theorem on the equivalence of separation and optimization: an optimization problem can be solved in a polynomial time if and only if there is a polynomial time separation ...
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