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

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Approximation ratio of a greedy grid-cover algorithm

We're given a $N\times M$ grid, and we want to cover all coordinates in the greedy by rectangles of size $\le k$. Consider the following greedy algorithm. At each iteration, it chooses a rectangle ...
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1answer
79 views

What is known about this TSP variant? [closed]

This question was cross-posted to cstheory.SE here. Imagine you're a very successful travelling salesman with clients all over the country. To speed up shipping, you've developed a fleet of ...
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0answers
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A query regarding the Max3SAT Approximation Algorithm's Application

Its known that a polynomial time Approximation Algorithm that satisfies 3MaxSAT in 7/8+e clauses implies P=NP. If given that the given 3MaxSAT is satisfiable, it is still difficult to find a 7/8+e ...
2
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1answer
30 views

Is it possible that low-resource Turing Machines can always “usually” agree with high-resource Turing Machines

Say that a language $L$ is a $f$-approximation of a language $L'$ if, for all input lengths $n$, $L$ and $L'$ agree on at least a fraction $f$ of the inputs. It is known that there are problems in ...
4
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1answer
55 views

Which bound is better, a logarithmic or a polynomial with arbitrarily small degree?

I have a randomized approximation algorithm which can be tuned by selecting the randomization probabilities. I found out that: For any $\epsilon >0$, there are probabilities for which the ...
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1answer
32 views

Knapsack Greedy Approximation: Worst Case

I am currently studying approximation algorithms and I have run into an issue with a study problem. The approximation algorithm is for the general Knapsack problem, and it proposes a greedy approach, ...
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1answer
30 views

Maximizing the minimum of $k$ submodular functions

Let $f(X)$ be a monotone submodular function from $2^{\{1, \ldots, n\}}$ to $\mathbb{N}$ and consider the problem of maximizing it subject to a cardinality constraint $|X| \leq m$. By using the greedy ...
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1answer
129 views

Approximating the diameter of graph G

Anyone has an idea how to solve this problem: Let G be an undirected, unit-weighted connected graph. Design a linear-time algorithm to obtain a 2-approximation of the diameter of G. I.e., the largest ...
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From hypercontractivity norm bounds to small set expansion property

Consider these two theorems on this theme, Lemma 8 on page 10, http://www.boazbarak.org/sos/files/lec2d.pdf Lemma B.1 on page 63 of http://arxiv.org/pdf/1205.4484v3.pdf Aren't these two theorems ...
0
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1answer
15 views

Is there an implicit condition OPT<P(|input|) in approximation schemes?

Sorry if my question is banal. Consider an approximation scheme such as FPTAS that guarantees to find a soln>(1-eps)OPT for ...
3
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1answer
57 views

Greedy algorithm for Set Cover problem - need help with approximation

I want to approximate how close is the greedy algorithm to the optimal solution for the Set Cover Problem, which I'm sure most of you are familiar with, but just in case, you can visit the link above. ...
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0answers
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FPTAS for knapsack with private valuations

The Knapsack problem has a well-known FPTAS based on rounding and then using the pseudopolynomial dynamic programming algorithm. When the valuations $v_i$ are private, we also need the assignment rule ...
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0answers
52 views

How to prevent overflow and underflow in the Euclidean distance and Mahalanobis distance

I was working in my project when I was struck by the question of whether it would be necessary, or at least cautious, prevent overflow and underflow in the calculation of these two distances. I ...
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3answers
1k views

Why are NP-complete problems so different in terms of their approximation?

I'd like to begin the question by saying I'm a programmer, and I don't have a lot of background in complexity theory. One thing that I've noticed is that while many problems are NP-complete, when ...
6
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1answer
58 views

Mathematical optimization on a noisy function

Let $f:\mathbb{R}^d \to \mathbb{R}$ be a function that is fairly nice (e.g., continuous, differentiable, not too many local maxima, maybe concave, etc.). I want to find a maxima of $f$: a value $x ...
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0answers
30 views

Reference for approximation algorithms [closed]

what is the best book to gain an introductory understanding of approximation algorithms? I'm looking for something along the lines of the Sedgewick, that has examples written in a well known language ...
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1answer
41 views

Perturbation of a vector

Given a vector $x=(x_1,\cdots,x_n)$ such that $0\leq x_i \leq1$ and $\sum_{i=1}^n x_i=1$. I would like to find a vector $x^*$ such that ($l_1$ norm ) $||x-x^*||_1\leq \delta$, where $\delta >0$. ...
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2answers
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Is this combinatorial optimisation problem similar to any known problem?

The problem is as follows: We have a two dimensional array/grid of numbers, each representing some "benefit" or "profit." We also have two fixed integers $w$ and $h$ (for "width" and "height".) And a ...
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1answer
48 views

Sorted-greedy for Load Balancing Problem

In load balancing problem we have $m$ machines and $n$ jobs, each taking processing time $t_j$. Total processing time on the machine $i$ is $T_i =\sum_{j\in A(i)}{t_j}$, where $A(i)$ is the set of ...
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3answers
46 views

Can approximation help find the exact answer?

Lets assume we have an array with 100 numbers and we want to find how many '1's there are. Best solution will be reading every numbers and counting. Now we get a hint that there are 50,51 or 52 '1' in ...
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1answer
72 views

Anagrams solver based on transitions probability

I have an English dictionary (text file) and the frequency of 2-grams, 3-grams and 4-grams as the beginning of each word. I need to write an algorithm that, with a given word, calculates the possible ...
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1answer
66 views

Subset-sum approximation algorithm running time

in 35.5 of CLRS i have read about algorithm to find sum as large as possible, but not larger than $t.$ Essential part of this algorithm is trimming. On every step you delete all numbers which close ...
1
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1answer
152 views

3/2 - Approximation probabilistic algorithm for MAX-3-COLOR

I have a textbook question here regarding Max-3-Coloring and need some assistance with it. I have searched for any type of information regarding it but haven't found anything substantial. Here is the ...
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0answers
17 views

MaxSNP flow problems

Currently, I'm trying to understand the definition and notion of MaxSNP and MaxSNP-hardness. I see that several combinatorical problems such as Max-3SAT are in MaxSNP since one can easily express them ...
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1answer
86 views

Approximate the order of summation of two vectors

Assume we are given two vectors $A,B$ that each contains $c$ numbers: $A=[a_i>0]_{1 \times c}$. We want to see the weighted summation of which one is larger. In other words, given two vectors $A$ ...
3
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1answer
36 views

What does the 2 in a 2-approximation algorithm mean?

Does the 2 in a 2-approximation algorithm mean the solution is within 2*OPT or OPT/2?
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1answer
131 views

Balanced Weight Distribution in Bins/Buckets

Let $W = \{w_1,w_2,...w_n\}$ be a set of integer weights. Let $B = \{b_1,b_2,...b_m\}$ be a set of buckets, with $m \leq n$. Let $T(b_j)$ represent the total weight present in bucket $b_j$, which ...
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0answers
43 views

Approximation for fewest incompatibilities in a scheduling algorithm

Suppose you have a task selection algorithm to select the largest subset of tasks that do no overlap. The greedy algorithm that selects tasks based on their finish time will always produce an optimal ...
3
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1answer
100 views

Does a greedy task selection algorithm find a c-approximate solution?

I was told this question may be better suited here. A scheduling problem can be stated as: Given a set $\{(s_i,f_i)\}_{1\le i\le n}\}$ of tasks identified by their start and end times, choose ...
2
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1answer
25 views

Are there FPTASs for the min cost flow problem?

In literature, one can find many approximation algorithms for the multicommodity min cost flow problem or other variants of the standard single-commodity min cost flow problem. But are there FPTASs ...
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2answers
104 views

A basic question about approximation algorithms for the Traveling Salesman Problem

Approximating the traveling salesman problem (TSP) within a constant factor $k$ is hard. The standard proof shows that the existence of such an approximation allows the Hamilton Cycle problem to be ...
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1answer
41 views

Proving approximation ratio

We recently in computational complexity class dealt with approximation algorithms and I was wondering how one would prove a heuristic having a certain ratio in regards to the optimal version. Looking ...
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1answer
27 views

Hardness and approximation of a problem with a parameter

Let $H$ be a decision problem, where we are given an integer $k$ and some object, say a graph or a formula. We know that $H$ is NP-complete for $k \geq c$, where $c$ is some constant like 3 ($H$ could ...
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2answers
37 views

What all can a valid approximation algorithm access?

For example can an approximation algorithm call a subroutine which is solving a NP-Hard problem? (like say its trying to find the longest path in some graph as an intermediate step) Is that allowed?
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smaller size approximation to minimum vertex cover

Does there exist a simple approximation to the minimum vertex cover problem that aims to find a smaller (or equal) set to the minimum? Usual algorithms seems to aim to find an approximation such that ...
2
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1answer
60 views

Why is Ibarra Kim for 0/1 knapsack an fully polynomial time approximation scheme (FPTAS)?

According to one of my CS lectures, there is an fully polynomial time approximation scheme for the 0/1 Knapsack problem. A first version was developed by Ibarra and Kim, but there are several improved ...
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3answers
712 views

Algorithm to distribute items “evenly”

I'm searching for an algorithm to distribute values from a list so that the resulting list is as "balanced" or "evenly distributed" as possible (in quotes because I'm not sure these are the best ways ...
4
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1answer
42 views

Probabilistic hardness of approximation or solution of NP-hard optimization problems under a probabilistic generative model for input data

So in biology (DNA sequences), sequence alignment is a generalization of longest common subsequence where an alignment of two sequences is scored typically with a linear function of how many spaces ...
6
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1answer
86 views

Does $\#W$[1]-hardness imply approximation hardness?

Let $\Pi$ be a parametrized counting problem, where the parameter is the solution cost, e.g. counting the number of $k$-sized vertex cover in a graph, parametrized by $k$. Assume that $\Pi$ is ...
11
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2answers
115 views

Does #$P$-Completeness imply approximation hardness?

Let $\Pi$ be some counting problem which is known to be #$P$-Complete. Does it imply that $\Pi$ is $APX$-hard (i.e. no PTAS for the problem exists unless $P=NP$)?
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Finding the upper bound of the length of a closed walk

I am having trouble understanding a part of the proof of Lemma 2 (Page 184). It says the length of the tour is $$ \leq \lceil n^{1/2} \rceil + \triangle(n + \lceil n^{1/2} \rceil) + \sqrt{2} $$ I ...
3
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1answer
28 views

Packing unsplittable flows problem

For a single stream of elements as input every elements should be routed into a fixed number of $k$ output streams trying to keep them balanced. In the following example $k=3$ : Let's define as ...
3
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1answer
26 views

What is the significance of the vector dimension in semidefinite programming relaxations?

Let's say that we want to design a semi-definite programming approximation for an optimization problem such as MAX-CUT or MAX-SAT or what have you. So, we first write down an integer quadratic ...
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1answer
286 views

What is a bicriteria approximation algorithm?

What is a bicriteria approximation algorithm? This keeps coming up in the case of data stream clustering. Is this related to multi-objective optimization? This is where I came across it: ...
3
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1answer
55 views

Hardness of approximating hitting set

Consider the hitting set problem with $n$ elements and $m$ sets. I gather from the linked page as well as this that 1) it is NP-hard to approximate the cost of the optimal solution to a ...
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0answers
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Approximation scheme for finding best product of matrices that minimizes $||Ax - y||$ for given $x,y$

Given a set of $N$ $n \times n$ matrices $A_1,\ldots,A_N$, and two vectors $x,y$, the problem is to find a product of up to $K$ matrices $A = A_{j_1}A_{j_2}\cdots A_{j_k}$ so that $Ax$ is as close to ...
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What is the Unique Games Conjecture? [closed]

What is the unique game conjecture in relatively simple words? What are the consequences of proving it or disproving it? Does it has any relation to game theory? Why is there "game" in the name?
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1answer
19 views

Estimating (c-1) from approximation of c

If we have a FPRAS for approximating the quantity c, can we get another FPRAS for estimating (c-1) using the estimation of c?
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1answer
144 views

Approximation algorithms for Euclidean Traveling Salesman [closed]

I am trying to find a way to solve Euclidean TSP in a polynomial time. I looked at some papers but I couldn't decide which one is better. What is the general approximation algorithm for solving this ...
0
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1answer
44 views

Can someone interpret what this is asking for

I have this programming problem, but I really cant figure out what it wants me to do. Heres what it is: The cube root of a number can be found based on the observation that, if $t$ is an ...