# Tag Info

## Hot answers tagged approximation-algorithms

Accepted

### Proof assistant usage in complexity theory research?

A general rule of thumb is that the more abstract/exotic the mathematics you want to mechanise, the easier it gets. Conversely, the more concrete/familiar the mathematics is, the harder it will be. So ...
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### Is the following graph optimization problem approximable within a constant factor?

The problem is very similar to Min Uncut. In Min Uncut, given a graph $G = (V, E)$, we need to find a subset of edges $E'$ s.t. $G - E'$ is bipartite; the objective is to minimize the size of $|E'|$. ...
• 3,919
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### Can we approximate the number of words accepted by an NFA?

There exists a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the problem of counting the words of length $n$ accepted by a NFA in the general case (without restricting to the acyclic ...
• 561
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### Does Max Planar 3-SAT admit a PTAS?

Yes, a PTAS for Max-Planar-3-SAT can be constructed by using Brenda Baker's approach. This has been observed, for instance, in Theorem 17 in Pierluigi Crescenzi and LucaTrevisan: "Max NP-...
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### Additive versus multiplicative accuracy

Why is it sometimes easier to achieve $\epsilon$-additive accuracy than $\epsilon$-multiplicative? Consider a problem where the objective value OPT is guaranteed to lie in a constant non-negative ...
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### Is the current best approximation ratio for Vertex Cover problem also a lower bound?

Cormen, Leiserson, Rivest and Stein say that this algorithm that achieves ratio of 2 is tight. They did not exclude the possibility of another algorithm achieving better. Vertex Cover is NP-hard to ...
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### Find research partner (profession and beginner)

I don't know of such a page. Most researchers have specific problems that they are interested in working on, and would only want to collaborate on those. If you pick a particular area and focus on ...
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### W[1]-hard problems with FPT time approximation algorithms

In the Directed Odd Cycle Transversal problem the input is a graph $G$ and the task is to find a smallest set $S$ of vertices such that $G-S$ has no (directed) cycles of odd length. In the ...
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### Why is the Greedy Conjecture so difficult?

Let me first try to summarize what is known about the Greedy Conjecture. Blum, Jiang, Li, Tromp, Yannakakis prove that the Greedy Algorithm gives a 4-approximation, and Kaplan and Shafrir show that ...
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### What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

I think for getting 1.99-approximation algorithm this paper by Manurangsi and Trevisan, has the current fastest algorithm.
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### Proof assistant usage in complexity theory research?

One very prominent example is of course Gonthiers Coq formalization of the 4 color theorem in Coq which uses a lot of combinatorics. My colleague Uli Schöpp used the ssreflect library developed by ...

### Is there a sensible notion of an approximation algorithm for an undecidable problem?

This is answering the title of the question more than its content, but you can also consider "approximations" of the halting problem as algorithms which will give you a correct answer on "almost all" ...
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### W[1]-hard problems with FPT time approximation algorithms

In [1], the authors prove that MaxSAT parametrized by the clique-width (resp. neighbor diversity) of the incidence graph of the CNF formula has an FPT-AS (Fixed Parameter Tractable Approximation ...
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### W[1]-hard problems with FPT time approximation algorithms

In Defective Coloring we are given a graph $G$ and an integer $\Delta^*$ and are asked to partition the vertices of $G$ into the minimum possible number of color classes so that each class induces a ...
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### Count satisfying assignments of CNF formulas over all possible negation assignments

The quantity $\sum_k|\phi_k|$ can be computed in polynomial time, in fact, in uniform $\mathrm{TC}^0$. By double counting, we have \sum_k|\phi_k|=|\{(a,k):a\models\phi_k\}|=\sum_{a\in\{0,1\}^n}|\{k:...
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### Is an algorithm with an approximation factor of 4000 useful?

A very good question! While I think a feasible solution which is 4000 times worse than the optimum is often not very practical, if you have several approximation algorithms to choose from, I would ...
• 6,665
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### kmeans++ for arbitrary metric spaces and general potential function

Here's an example that suggests that the stronger claim in the earlier version (Lemma 5.3) is false. I've only given a cursory look at the papers, so please check this carefully to make sure I am ...
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### Can we approximate the number of words accepted by an NFA?

And now there is a faster FPRAS: https://arxiv.org/abs/2312.13320
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### W[1]-hard problems with FPT time approximation algorithms

The k-cut problem is to remove a minimum number of edges to create at least k components. W[1] hard when parameterized by k but admits a 2-approximation for any k.
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### W[1]-hard problems with FPT time approximation algorithms

(This question was asked two years ago, but I'll post the answer for other people who may see this question.) In the Capacitated $k$-median problem we are given a set $F$ of facilities, each facility ...

### Max cut problem between two connected subgraphs

Here is a straightforward reduction from the max-cut problem: Take any graph and add two new vertices $u,v$ and connect them to every other vertex with weight 0 and connect them to each other by a ...
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