Questions tagged [approximation]

Questions about algorithms that solve problems up to some bounded error.

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Piecewise linear approximation for set of points

If we have a set of points $A$ and a known algorithm $Bestfit(A)$ to find the best-fit straight line-segment through $A$, what would be a good algorithm to construct piece-wise linear segments (with ...
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2answers
42 views

Numerical Approximation in Java

I am trying to solve an equation which I believe cannot be done analytically, but can use a numerical approximation to get a result. The equation is: $$\frac{2*\sqrt{\pi}*h*s*e^{m^{2}/(2*s^2)}}{\sqrt{...
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How to prove that degree-4-IS si NP-Complete

I would like to have an idea to solve this kind of problem. Let us call degree-k-IS the restriction of the problem of the independent set to graphs of degree maximum $k$. Show that the degree-4-IS ...
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32 views

Approximation Set Covering Problem

I am studying approximation Set Covering problem, from "Introduction to Algorithms by Thomas H. Cormen" book. What I cannot understand is why they use harmonic numbers to the proof? What harmonic ...
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1answer
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3-Approximation for General position subset

I am currently studying for an exam and stumbled upon the following task: Given the following problem: Input A set of points $P \subseteq \mathcal{Q}^2$ and $k \in \mathbb{N}$ Question Find the ...
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Minimum-cut with balanced and limited number of nodes in each partition: Does this have an efficient solution or even a name?

I'd like to remove the minimum number of edges from an undirected unweighted graph to partition the nodes into an arbitrary number of connected components $S_1$, $S_2$,$S_3$,... $S_k$ while maximizing ...
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86 views

PTAS for Multiple Knapsack with Uniform Capacities, fixed number of Knapsacks

Consider the following problem: We are given a collection of $n$ items $I = \{1,...n\}$, each item has a size $0 < s_i \le 1 $ and a profit $ p_i > 0 $. There are $m$ (a fixed number) of unit-...
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Scheduling jobs on a single machine - minimising the weighted sum of completion times

Consider the following problem: there are $n$ jobs $\{1,...,n\}$, each has a processing time, $p_i$, a weight $w_i$, and an arriving time $r_i$. The goal is to minimise the weighted sum of completion ...
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Graham 1966 Minimum makespan scheduling

I am trying to understand how Graham's 1966 makespan algorithm results in 2-approximation. I have searched in google and I have found files, and videos on youtube about makespan scheduling, but again ...
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1answer
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Compendium of approximation ratios, with narrower scope and more up to date than the NP-compendium?

I usually check the NP-compendium from Pierluigi Crescenzi, and Viggo Kann when I want to know the APX status and approximability results of a problem. However, I understand that maintaining it is a ...
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Calculating match % and ranking according to that

I'm creating a website like where users will answer some yes/no questions set by me, up to them how many of those questions they want to answer. After a user submits his answer(s), he will be shown ...
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A variant of the knapsack problem

Consider the following variant for the knapsack problem: the input are disjoint sets of items $ T_1, T_2, ..., T_m$ (each contains items of a different type). Every item $i$ has a value of $v_i$ and a ...
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Compute the expected size of an approximation of vertex cover

Consider the following randomized approximation algorithm of vertex cover: Input: A graph G = (V, E). Output: A set $C_G \subseteq V$ a vertex cover of $G$. The algorithm: Set $C_G := \emptyset$. ...
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FPTAS algorithm to find flow at each link for multi commodity flow problem?

Given a graph $G$ and $K$ commodities to route from source to destination. I want to find, what is the maximum beneficial flow for each of the commodities and the relevant paths. I understand the ...
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Derandomization of vertex cover algorithm

I have the following randomized-algorithm for the vertex cover problem. Let $B_0$ be the output set: Fix some order $e_1, e_2,...,e_m$ over all edges in the edge set E of G, and set $B_0=∅$. Add to ...
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1answer
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Integrality gap and LP-rounding

I have a doubt about integrality gap. If I know that there is no integrality gap for a given problem, i.e.: $$\frac{\mathrm{OPT}(\mathrm{ILP})}{\mathrm{OPT}(\mathrm{LP})} = 1 \text{ (right?)},$$ ...
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Estimating number of points in 1D space

There are some arbitrary-chosen points in 1D space. What needs to be found is the approximate number of them without counting all of them. It is possible to choose some coordinates (numbers) and for ...
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Doubt on integrality gap and LP relaxation

I have an exercise that tells me that, given a problem P (of which now I omit the description) there is no integrality gap between LP and ILP formulation of this problem, and for every fractional LP-...
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176 views

2D interval scheduling problem

Suppose I give you $n$ axis-aligned rectangles with a specified width, height, and x-position (of the left edge) $\{(w_i, h_i, x_i) \mid i \in \{0, \ldots, n - 1\}\}$, as well as a bound $(y_\mathrm{...
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Prove that the 2-approximation of a modified local search algorithm for max-cut is tight

Consider the following local search approximation algorithm for the unweighted max cut problem: start with an arbitrary partition of the vertices of the given graph $G = (V,E) $, and as long as you ...
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ln(n) + 1 Approximation for Set Cover constructions

Set Cover Problem: Given a set $X$ and a collection of subsets $S_1, S_2, \ldots S_m \subseteq S$, we want to find the smallest cardinality of a set of $k$ elements $\{i_1, \ldots i_k \}$ such that $\...
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1answer
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Minimum edge weight k-exact cover with triangle inequality

Suppose you have a weighted graph $G = (V, E \subseteq V^2, w \in E \to \mathbb{R}^+)$, where $w$ satisfies the triangle inequality $w(x, y) + w(y, z) \ge w(x, z)$. Suppose $k \in \mathbb{N}$ (in ...
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Approximation algorithms for indefinite quadratic form maximization with linear constraints

Consider the following program: \begin{align} \max_x ~& x^TQx \\ \mbox{s.t.} ~& Ax \geq b \end{align} where $Q$ is a symmetric (possibly indefinite) matrix and the inequality is element-wise ...
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A heuristic for finding an edge cycle cover

I am looking to find a minimum list of cycles in a graph such that their union gives the list of all simple cycles in this graph. In the example below, here are 4 simple undirected cycles: 1-2-3, 2-3-...
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Tight analysis for the ration of $1-\frac{1}{e}$ in the unweighted maximum coverage problem

The unweighted maximum coverage problem is defined as follows: Instance: A set $E = \{e_1,...,e_n\}$ and $m$ subsets of $E$, $S = \{S_1,...,S_m\}$. Objective: find a subset $S' \subseteq S$ such ...
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1answer
29 views

Reductions from non decision problems

I want to show a minimization problem $Y$ has no approximation factor of 1.36. To be more specific the problem $Y$ is the exemplar distance problem between two genomes. Could I reduce from the min ...
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Approximate algorithms for class P problems

As a part of my Algorithm course we studied Approximate Algorithms for NP-complete or NP-hard problems, e.g. "set cover", "vertex cover", "load balancing", etc. My professor asked us as an extra ...
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Rearrange items in order reduce fragmentation and reduce wasted space

I have a segment with some offsets at irregular intervals There are items of various length inside. Items cannot be placed randomly. Instead, their left side must match some offset. Items are free ...
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What are some counter examples for Load balancing problem?

I learnt about load balancing problem under approximate algorithms. I am learning about it, and studying methods to counter part it's non -solvability under P time. I am quite out of my examples, and ...
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Randomized version of the class $APX$?

Is there a class which is to APX what BPP is to P? I'm looking for a definition that is like the following: "For $r > 0$, an $r$-RPCA (randomized polynomial-time constant-factor approximation) ...
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1answer
80 views

Optimal solution for Weighted points problem

Problem: Fix a constant $k$. Given a set of $2d$-dimensional points $N = \{N_1, N_2, N_3, \dots, N_n\}$, each associated with an arbitrary weight, find a set of points $X = \{X_1, X_2, X_3, \dots, ...
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About Steiner tree problem in graphs

In the paper (p. 3) and the slides presents the formulation of the Steiner problem on graphs via so called Steiner cuts. But according to the definition, the number of Steiner cuts and so the ...
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2answers
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Can non-metric TSP be approximated within some non-constant value?

It is known that metric TSP can be approximated within some constant value, such as 3/2 through Christofides' algorithm. It is also known that non-metric TSP cannot be approximated within some ...
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39 views

Greedy algorithm for feedback vertex set / greedy algorithms vs local ratio in general

A greedy algorithm for finding a minimum feedback vertex set is to pick and remove a vertex with minimum $w(v)/\delta_H(v)$, where $H$ is the current graph, until there are no more cycles left. (...
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54 views

What is the approximation for odd cycle transversal?

What is the best approximation for odd cycle transversal? (on general graphs) Sorry if this is easily found everything I found about odd cycles is about paramaterized complexity and kernels
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Algorithmic question: distribute balls, optimise for balancing (i) weights (ii) probabilities of picking balls

I have an algorithmic problem that requires some lengthy explanation, which follows below. tl;dr: distribute balls with weights among bags, optimise for balancing both (i) the weights between the ...
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1answer
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How to use c-gap problems to prove inapproximability?

Suppose there is a specific set function with some properties - $f=2^V\to \mathcal{R}$. It is known that the following problem is NP-Hard: Find $S\subseteq V, |S|\leq k$ such that $f(S)$ is maximized....
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1answer
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Hardness of approximation statement clarification?

In the paper I'm reading, there is a hardness of approximation result for an algorithm proved using a reduction to set cover. Roughly, the claim states that if there existed an algorithm that solved ...
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1answer
117 views

Difference Between PTAS and FPTAS [duplicate]

According to this link: Polynomial Time Approximation Scheme (PTAS) is a type of approximate algorithms that provide user to control over accuracy which is a desirable feature. These algorithms ...
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1answer
53 views

Is this variation of set-cover NP-hard to approximate?

The classic set-cover problem is described as follows: Let $S = \{s_1, ..., s_n\}$ be a target set, and let $\Lambda = \{A_1, ..., A_m: A_i \subset S\}$ be a collection of subsets of $S$. The ...
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Peculiar MCMC sampling problem

I have two random variables, X and Y, and Y is a positive real number. I can sample from $p(y|x)$, but I need to sample from $p(x)$, which I know to be proportional to $\frac 1 {E[y|x]}$. I could ...
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57 views

Alternative criterion for approximate maximum-weight perfect matching algorithms [closed]

Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not the factor between the approximate and exact weight sum achieved by each solution, but ...
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1answer
61 views

Analysis of an approximation claim

Consider the load balancing problem on two machines. Thus we want to distribute a set of $n$ jobs with processing times $t_1,...,t_n$ over two machines such that the makespan (maximum of the ...
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An LP with two covering constraints - how to round

I came across an LP with two covering problems, and I wonder how to find a good approximation. For the relevant part of the LP: We have a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
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1answer
32 views

Non-existence of approximation algorithm for the knapsack problem

I am working on the following exercise: Prove that if $P \neq NP$, there cannot exist an approximation algorithm $A$ for the knapsack problem (KP) such that $\exists k \in \mathbb{N}, \forall I \in S: ...
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56 views

Randomized algorithm to compute cover radius?

I am self-study the book "Geometric Approximation Algorithms" by Sariel Har-Peled. And I stuck on a problem and don't know how to start it. Let $C$ and $P$ be two sets of point in the plane , such ...
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1answer
49 views

What are the current state of art approximation algorithm for NP-Hard problems? [closed]

I came cross some works try to use deep learning to approximate NP-Hard https://arxiv.org/pdf/1810.10659.pdf Though the paper seems to have very good results but based on the citations. I'm quit ...
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1answer
41 views

Shortest hamiltonian path for different dimension points

The shortest Hamiltonian path (solution) for a set of points in $\mathbb{R}^k$ (in Euclidean space) changes subject to $k$. For example if for $k=1$, the shortest Hamiltonian path will be the sorted ...
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1answer
55 views

What is an approximation factor for the Greedy Motif Search algorithm?

What is approximation factor for the Greedy Motif Search algorithm? I couldn't find an answer to my question except for the fact that the algorithm has a unknown aproximation factor. I'm not a native ...
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1answer
64 views

Is there an algorithm to overapproximate a context free grammar by a regular expression?

I understand that a context-free grammar is strictly powerful than a regular expression in that a context free grammar can represent any regular language, but not all context free languages can be ...

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