Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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92 views

What are the worst-case and average-case time complexities of the greedy algorithm for the weighted set cover problem?

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 $...
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Workshops/summer school for discrete optimization/discrete math [closed]

I found a list List of TCS conferences and workshops however, I'm looking for workshops and summerschools that can teach general techniques/ideas in discrete optimization to someone who is not well ...
3
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1answer
87 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-...
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Application of a Poisson inequality in the networks

I know that there are many works where Poisson (https://en.wikipedia.org/wiki/Poisson_distribution) processes are used in the construction of networks models. I am seeking for a model which can be ...
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1answer
327 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 ...
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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 ...
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1answer
69 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 ...
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33 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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1answer
94 views

Additive versus multiplicative accuracy

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 ...
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1answer
64 views

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 ...
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Polynomial time multiplicative approximation algorithms for logistic regression?

Typically algorithms for logistic regression have an iterative aspect since the problem does not admit a closed form solution. By extension, most iterative algorithms (gradient descent etc.) for ...
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33 views

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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1answer
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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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1answer
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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 ...
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1answer
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Finding the size $k$ subset in a metric space that maximizes the min distance between elements

I have a metric space $(X,d)$ and I'd like to find a subset of size k of far away elements. We can cast this as the following optimization problem $\max_{S \subseteq X, |S| = k} ( \min_{i \not = j, ...
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Complexity of a scheduling problem with a fixed left bound of jobs

Consider the following scheduling problem. We have a finite set of jobs $\mathcal{J}= \{j_1, j_2, ..., j_n\}$ to do. Every job $j_i$ has its own value $c_i$ (the amount of money we are paid for ...
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1answer
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is this selection problem np-hard? [closed]

Give $n$ clusters $C=\{C_i\}_{i=1}^n$ where each cluster consists of a set of similar points, i.e., $C_i=\{c_j\}_{j=1}^{|C_i|}$. The similarty between two points $c_i$ and $c_j$ is denoted as $w(c_i,...
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1answer
132 views

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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1answer
649 views

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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1answer
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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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1answer
951 views

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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Relaxed minimum dominating set

(I moved this question from cs exchange to here, because it might be more on the topic here) Let $G=(V,E)$ be a directed graph with $n$ nodes. For a subset of nodes $S\subseteq V$, let $\mathcal{N}(S,...
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What is the run-time of LP?

Are there any further generalizations known to the result about run-time of a LP than what is stated in Theorem 1 of these lecture notes, https://nisheethvishnoi.files.wordpress.com/2018/05/lecture71....
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Approximate nearest neighbor search for $(\ell_p)^p$ metric, $0 < p < 1$

It is well known that approximate bichromatic nearest-neighbor search can be solved in sub-quadratic time for the $\ell_1$ and $\ell_2$ norms, as well as some other metrics. Are there any such results ...
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1answer
360 views

Why is the Greedy Conjecture so difficult?

I recently learned about the Greedy conjecture for the Shortest Superstring Problem. In this problem, we are given a set of strings $s_1,\dots, s_n$ and we want to find the shortest superstring $s$ ...
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216 views

Martingale exit arguments for gradient Langevin dynamics

I am concerned about the proof of Lemma 6.3 (page 18) of this paper, https://arxiv.org/pdf/1902.08179.pdf which shows that for smooth convex functions the gradient Langevin dynamics has a high ...
6
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1answer
230 views

What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard. There is also a parameterized algorithm for ...
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79 views

Alternative criterion for approximate maximum-weight perfect matching algorithms

I have asked this question in cs.stackeschange, and it was recommended to me that I asked it here. Is there any literature on approximate maximum-weight perfect matchings where the approximation ...
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51 views

A “cut” packing problem

Consider the following maximisation problem. Given a path $P=\{1,2,\dots,n\}$ over $n$ vertices, a set $D\subseteq P\times P$ of "edges" and a set of positive integers capacities $\mathcal{C}=\{c_{i,...
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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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1answer
103 views

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 ...
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1answer
155 views

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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1answer
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NP-intermediate approximation regimes for natural problems within the MAX-k-CSP family

I would like to know whether there are any examples of natural problems within the MAX-$k$-CSP family for which (under standard/reasonable conjectures) we believe the following: There is a value $\...
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1answer
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How is additive error handled in this simple algorithm? 'Product of all elements'

Say we have two unit vectors $\hat{u}, \hat{v} \in \mathbb{R}^n$ where $\hat{u} = (u_1,...,u_n)$ and $\hat{v}$ approximates $\hat{u}$. $~\hat{v} = (u_1+\epsilon, ...,u_n+\epsilon)$ where $\epsilon = \...
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2answers
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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 ...
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Stable recovery of signals by $\ell_1$ optimization

Suppose the received vector $y$ is generated from a vector $x^*$ as $y = { D}x^* + z$ for some ``dictionary" matrix ${D}$ and noise vector $z$ s.t for some $\epsilon >0$ we have, $\Vert z \Vert_2 \...
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1answer
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Intuitive explanation behind Goemans-Williamson randomized rounding

A very simple randomized cut algorithm achieves $1/2$ of the optimal value: just choose each vertex to be in the cut with probability $1/2$, independently. Goemans-Williamson does something more ...
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What is the competitive ratio of a $d$-way associative LRU cache?

In a caching problem, items arrive online, and the algorithm needs to decide which elements to keep in the cache. If the current item is not cached, we pay a penalty of $1$. It is well known that for ...
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1answer
115 views

Maximum-minimum satisfiability

In MAX-SAT, given a formula, we want to maximize the number of satisfied clauses: given a formula $\phi = c_1 \cap \cdots \cap c_n$, where each $c_i$ is a disjunction, we want to find the largest $k\...
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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 [1] 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 ...
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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 $j$-...
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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 ...
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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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1answer
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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 ...

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