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### Minimum vertices to cover other vertices with max weight [duplicate]

I have a problem where I'm given the input of a graph. The output would be a set of vertices such that I have the minimum number of vertices to cover other vertices and if there is more than one ...
3k views

### Why there are no approximation algorithms for SAT and other decision problems?

I have an NP-complete decision problem. Given an instance of the problem, I would like to design an algorithm that outputs YES, if the problem is feasible, and, NO, otherwise. (Of course, if the ...
8k views

### What is practical difference between NP and PSPACE-complete?

Here's something that has puzzled me lately, and perhaps someone can explain what I'm missing. Problems in NP are those that can be solved on a NDTM in polynomial time. Now assuming P$\,\neq\,$NP, ...
842 views

### $1+\epsilon$ approximation for inapproximable problems

I am currently confused by the following situation: 1) The metric $k$-center problem is inapproximable in polynomial time within $2-\epsilon$ unless $P=NP$. 2) The metric $k$-center problem can ...
4k views

### Algorithm for finding maximum mutually coprime subset of a multiset of integers

For a certain problem I am trying to solve given a list of integers, it is advantageous to me to first identify as many of the integers that are mutually coprime as possible. I'm having trouble ...
4k views

### Weighted interval scheduling with m-machines

I am looking for some advice and direction on solving the weighted interval scheduling problem with $m$-machines to plan some experimental "wet lab" procedures. The problem is very similar to the ...
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### NP complete problems that are solvable in polynomial time if the input (e.g. number of variables) is fixed?

I have seen some problems that are NP-hard but polynomially solvable in fixed dimension. Examples, I think, are Knapsack that is polynomial time solvable if the number of items is fixed and Integer ...
639 views

### NP-complete decision problems - how close can we come to a solution?

After we prove that a certain optimization problem is NP-hard, the natural next step is to look for a polynomial algorithm that comes close to the optimal solution - preferrably with a constant ...
3k views

### Longest path in a cyclic, directed and weighted graph

I am looking for the longest simple path in a directed, cyclic and weighted graph with positive and negative weights. In my research so far I have found out that you need to generate ...
295 views

### How can you bound the error of an approximation without knowing the optimal solution?

I been looking at this site and it says that people found solutions for TSP tours that are just 0.031% higher than the optimal tour is. Without finding the optimal tour how does they know what length ...
1k views

### Cover points with minimal number of spheres of fixed radius

I have a set of k n-dimensional points: P1(x11, x12, ..., x1n), P2(x21, x22, ..., x2n), ..., Pk(xk1, xk2, ..., xkn). A distance D(Pa, Pb) is defined between any two points, which satisfy usual ...
392 views

### Pick a subgraph that maximizes the total cost of min-spanning tree among all subgraphs of the same size

There is a complete graph $G$ with $n$ vertices and each edge has a distinct weight. Is there an efficient (not necessarily optimal) algorithm to select $k$ vertices from the graph $G$, such that the ...
129 views

### How can I evaluate an algorithm for a NP-Hard problem?

I have written a program to calculate the number of stable partition in a graph. ( That is: find which partition of the nodes does not have edges between nodes of the same block. ) The professor, ...
276 views

### Why does the NP completeness of the Hartree-Fock method not lead to difficulty in practical calculation?

I read Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory. In appendix, it is claimed that In the following, we show that approximating the ...
733 views

### Searching the best trading route - algorithm

Imagine a village with people trading goods. Each person has his own offer in this format: Giving amount a of good b for amount <...

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