# Questions tagged [approximation-algorithms]

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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 ...
1 vote
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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 ...
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### Complexity of approximating a real function using queries

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 ...
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Let $X$ be the universe of elements, $F$ a collection of subsets $S \subseteq X$, each with an associated cost. The goal is to find a subcollection $C \subseteq F$ of minimum total cost which covers $... 6 votes 1 answer 296 views ### Good Survey paper for k-means/k-median/k-center/facility-location I have stated 4 problems in the Question title. All these problems are closely related and are studied in various variations. For example: Space: Euclidean/metric/discrete/continuous/non-metric/2-... 10 votes 1 answer 610 views ### Is the following graph optimization problem approximable within a constant factor? Let$G=(V,E)$be an undirected graph, and let$\pi$be a permutation of the vertices in$V$. For a node$v\in V$, we denote by$\text{pred}_{\pi}(v)$(respectively$\text{succ}_{\pi}(v)$) the set of ... 1 vote 0 answers 178 views ### Separation oracle for hitting all small cut on a graph? We are given as input an undirected graph$G=(V,E)$, weights$w_e \ge 0$for all$e\in E$and an positive integer$k$. We aim to select a set of edges with the minimum weight, such that the cut set of ... 5 votes 2 answers 244 views ### kmeans++ for arbitrary metric spaces and general potential function I was reading this popular paper "k-means++: The Advantages of Careful Seeding". It appeared in SODA 2007. Since this technique is the most popular clustering technique, I am hoping that my question ... 0 votes 0 answers 40 views ### Different version of approximation complexity and algorithm for densest-k-subgraph problem In the densest-k-subgraph problem, we are given a graph$G =(V,E)$and$k$, and we are asked to find a set$S \in V$of vertices to maximize the number of edges in the induced graph of$S$, i.e.$|Ind[...
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I am trying to understand the difference between $\epsilon$-additive and $\epsilon$-multiplicative algorithms. The way I understand this definition is as follows. An $\epsilon$-additive algorithm is ...
1 vote
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### How can I find dependent rounding procedures with the desirable properties?

I'm seeking for materials on dependent rounding. However, what I've found are two papers: Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A., 2006. Dependent rounding and its applications to ...
1 vote
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### Efficient algorithm for finding segregators in a directed acyclic graph

Given a directed acyclic graph $G=(V,E)$, we define a $(\alpha,\beta)$-segregator of $G$ to be a subset $S$ of $V$ of size $\alpha$ such that no vertex in $G\setminus S$ has more than $\beta$ ...
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### Approximation Ratio of Local search for $k-$center problem

In the $k-$center problem, you're given $V$ points in Eucledian space, and you're asked to get a subset $C\subset V, |C|=k$ such that $\max _{v\in V}d(v, Closest-Center(C,v))$ is minimized. Now I am ...
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### Is it possible to approximate the solution of NP-Hard problems in polynomial time using linear programming? [closed]

Suppose we have a NP-Hard problem such as the k-col, which is meant to determine if a graph may be colored using at most ...
1 vote
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### Is this greedy algorithm for vertex cover studied before?

For the vertex cover problem (Given a graph $G$, find a minimum set $S$ of vertices such that $G - S$ is edgeless), there are two famous greedy algorithms. The edge greedy (finding an edge and taking ...
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### Sequential vs Distributed algo question

If a certain graph problem in the $\textbf{sequential}$ setting is proven to have "no" better constant-factor approximation algorithm than say a 2-approx. algorithm in polynomial time, then does this ...
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### the shorstest cycle containing two given points

I am given a edge-weighted (multi)graph $G$ and two of its vertices, $u, v\in V(G)$. I want to find two edge-disjoint paths that connects $u$ and $v$ while minimizing the sum of the lengths of the ...
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### Is an algorithm with an approximation factor of 4000 useful?

A paper published in SODA this year (2019) proposed a constant approximation algorithm for the lower bounded facility location problem with general lower bounds. To my surprise, when reading the ...
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### What is the communication complexity of approximating addition?

In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. Specifically, the class of problems I am interested in is parametrized by four ...
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### finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
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### Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
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### Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
1 vote
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### Count satisfying assignments of CNF formulas over all possible negation assignments

Consider the set of all CNF instances that can be generated by adding negations to a single monotone CNF instance. How hard is it to compute the sum of the counts of satisfying assignments for the set?...
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### $XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?

Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following: $FPT = XPuniform\implies EPTAS = PTAS$. Fundamentals of Parametrized complexity on page $534$ has ...
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### Machine Learning Algorithm To Fill Data Holes

I'm having trouble finding a good place to begin with this. I'm just looking for a name or point to start researching a Let's say I have 1000 records. 10 of these records are only 90% complete. The ...
1 vote
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### Minimum relevant variables in linear system - additive approximation

In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.: $$A x = b$$ and the goal is to find a solution $x$ with as few nonzero variables as ...
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### Best polynomial-time approximation factor for NP-optimization problems

Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold: There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
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### Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour  states that if graph G has large treewidth, then it contains a large grid as minor. Most approximation results on general classes of graphs with ... 1 vote
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### PTAS for projective clustering : survey

$(k,j)$-projective clustering is the natural generalisation for k-clustering, in which one needs to find $k$ $j$-flats in $\mathbb{R}^d$ that minimizes the cost function as defined below: Given a $j$-...
1 vote
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### Best approach for allocation problem

I am a bit rusty on optimization algorithms and need an advice. This is my problem: I have n images (with width and ...
1 vote
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### Why do k min-hashes, instead of one hash where we find the k minimum elements?

Traditionally if one wants to sketch streams for Jaccard similarity hashing, one finds the minimum element in each of $k$ permutation for comparison purposes, and then takes number_of_collisions / $k$ ...
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### Does Max Planar 3-SAT admit a PTAS?

Suppose we are given a formula $\phi$ of 3-SAT, with variables $x_1,\dots, x_n$ and clauses $C_1,\dots, C_m$. Consider the graph $G_\phi$ where there is one node for each clause $C_i$, for each ...
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### Classification of randomized approximation algorithms

Is there a known classification of randomized approximation algorithms, in the same vein as the distinction between Monte Carlo and Las Vegas algorithms for decision problems? (Or equivalently ...
We want to cover $n$ elements with some sets from $S_1, …, S_m$ (classical set cover). We furthermore suppose that any element belongs to at least $k$ sets and want to find a set cover with cardinal ...
### Counting the maximum number of paths of length $n$ that differ in at least $k$ edges
What is known about the complexity of solving (or approximately solving) the following problem? INPUT: Graph $G=(V,E)$ and constants $L$ and $K$. OUTPUT: The maximum size of any set $S$ of simple ...