# Questions tagged [approximation-algorithms]

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### Is there a primal-dual algorithm for the Tree Augmentation Problem or the Cactus Augmentation Problem?

The TAP problem and the CacAP problem can be seen as covering problems for the minimum cuts of a graph. It seems like these problems would fall under the framework of network design problems (...
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### k-Median Problem With Restricted Centers

The $k$-median problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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### State of the art approximation algorithm for $\text{MAXCUT}$ that does better than Goemans and Williamson

I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia: The polynomial-time approximation algorithm for Max-Cut with the best known ...
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### A Special case of Weighted Set Cover Problem

We are given a set P of k points in the plane, and a set S of simple polygons that are triangulated into a set T of triangles, each triangle t in s has a weight that is 1 if we choose no triangles (w(...
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### Failing to understand a lemma regarding Robust Low Rank Approximation

I am reading Low Rank Approximation in the Presence of Outliers by Bhaskara and Kumar and kind of stuck at the proof of Lemma 9. The paper studies robust (to outliers) low rank approximation problem. ...
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### Modifying sets to minimize the distance among each pair of the mean value of sets

Given $n$ points, each point $x_i$ has a value $v_i \in \mathbb{R}^{d}$, and there are $m$ point sets $\{S_1,\dots, S_m\}$ that each point set consists of some points. The size of point sets can be ...
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### Low Rank Approximation of a hidden subset

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $Q\subseteq P$ with $\vert Q\vert \geq \alpha n$ for some constant $\alpha\in(0,1]$. Given a $j$-dimensional affine subspace(flat) $F$ consider the ...
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### Set cover where consecutive sets differ by at most one item [closed]

First I define my version of the set cover problem: We have a collection of sets such as $S_1, \dots, S_m$ where each $S_i$ is a subset of $M=\{1,\dots, m\}$. The goal is to find the minimum number of ...
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### Hardness of Approximation of Continuous Metric k-Median

First let me describe the metric $k$-median problem. Definition (Metric $k$-Median): Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ ...
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### Balanced and general $MAXkSAT$ known approximation results and bounds from $UGC$

$MAX2SAT$ has a $0.9401$ to $0.9402$ approximation algorithm which is conjectured to be optimal by $UGC$ while there is a balanced $MAX2SAT$ bound of $0.943$ approximation which is conjectured to be ...
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### A k-approximation to k-way number partitioning

The $k$-way number partitioning problem accepts as input a multiset $S$ of positive numbers, and returns a partition of $S$ into $k$ subsets such that the subset sums are as nearly-equal as possible, ...
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### Is this homework problem on T-joins wrong? [closed]

In Question 9.3a, it states that if $T=V$, then the minimum cost perfect matching is the minimum cost T-join. Is this actually true? I think I have a counterexample which I have drawn below.
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### A continuum version of the 1D k-means clustering problem: constant factor approximations

Modify the k-means clustering problem in 1D by assuming that, instead of a finite number of observations, we must classify into $K$ clusters a continuum of observations, distributed on the unit ...
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### Best approximations of Minimum Dominating Sets in chordal graphs

I am searching results and papers related to the (in)approximability of the Minimum Dominating Set problem in chordal graphs. In particular, what is the best approximation ratio achievable in polytime ...
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### Algorithms and approximations for optimal offline binary tree operations

Let's say we are using a binary tree to represent a set of elements, with operations $\mathsf{insert}(x)$ and $\mathsf{delete}(x)$. We will assume that the operations are used such that a deleted ...
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### Can hash functions speed up quantum simulation? (Generalizing May and Schlieper's idea) [closed]

To conform with the CS Theory SE crossposting rules, I've created a separate post for dequantizing Shor's algorithm (discussion on the Quantum Computing Stack Exchange was mostly about Shor's ...
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### Jump number approximation algorithm

A linear extension $x_1 x_2 \ldots x_n$ of a partially ordered set (poset) is said to have $k$ jumps if there are $k$ occurrences of consecutive elements that are incomparable with each other -- i.e., ...
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### Can the theory of Bidimensionality be applied to weighted instances of a problem?

So my understanding of bidimensionality is you are assured the problem solution is about O(k^2) so you can pay O(k) purely to reduce the instance to one of bounded treewidth. As far as I know, this ...
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### polytime approximability of directed multicut

Does anyone know how well directed multicut can be approximated in planar and minor free graphs? Also any survey of approximability of directed multicut and multicut in various graph classes would be ...
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### approximate maximum clique given vertex cover

I have a non optimal vertex cover of size k of a graph G, and I want to get a (1+epsilon)-approximation kernel of size linear in k for maximum clique of G. One thing I got is that every clique in G ...
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### Where to find info on (polytime) approximability of various discrete optimization problems?

Where to find info on (polytime) approximability of various discrete optimization problems? Sorry if this is stupid,but is there a site or reference that keeps up to date info on approximability of ...
Consider the following computational problem, where $I$ is the real interval $[-1,1]$: There is a monotonically-increasing function $f: I\to I$. You are allowed to access it only through queries of ...