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

learn more… | top users | synonyms (1)

5
votes
0answers
37 views

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 ...
0
votes
1answer
17 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 ...
-1
votes
3answers
43 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 ...
1
vote
1answer
46 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 ...
1
vote
1answer
18 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
vote
1answer
101 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 ...
1
vote
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 ...
0
votes
1answer
76 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
votes
1answer
23 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?
3
votes
1answer
46 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 ...
1
vote
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
votes
1answer
89 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
votes
1answer
20 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 ...
1
vote
2answers
72 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
20 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 ...
0
votes
2answers
35 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?
0
votes
0answers
44 views

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
votes
1answer
51 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 ...
11
votes
2answers
367 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
votes
1answer
30 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 ...
5
votes
0answers
51 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
votes
2answers
107 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 $P=NP$)?
1
vote
0answers
35 views

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
votes
1answer
26 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
votes
1answer
21 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 ...
1
vote
1answer
162 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
votes
1answer
42 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 ...
1
vote
0answers
16 views

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 ...
-1
votes
0answers
41 views

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?
1
vote
1answer
18 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?
1
vote
1answer
111 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
votes
1answer
42 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 ...
1
vote
1answer
55 views

Wave propagation in digital image

I believe the following question in summary is: How to approximate Euclidean distance in a digital plane? When a pebble falls on a calm surface of water a circular wave propagates. I want to color ...
1
vote
1answer
20 views

Error estimates of piecewise-linear curve approximations

In order to plot a curve a set of points is usually calculated based on some formula. The function FPLOT in MATLAB also supports plotting with some error tolerance. Its help says the following about ...
5
votes
1answer
157 views

How do GPUs compute sines?

I've been wondering lately how GPUs compute sines and cosines, and Google hasn't helped me finding a precise answer. Initially, I was thinking that in order to make the computations as fast as ...
3
votes
0answers
50 views

What functions are easy to optimize?

Say I have variables $w_1, \dots w_n, h_1, \dots h_m \in \mathbb R$, constants $W, H$, functions $f_1, \dots f_k : \mathbb R\times\mathbb R\to\mathbb R$ from some family $F$ and for each function ...
1
vote
1answer
80 views

Local search: Problem with neighborhood definition

I have question on understanding the following neighborhood relation within a local-search approximation scheme. Let $M$ be a legal matching on any bipartite graph. Let $U_k$ be the neighborhood ...
6
votes
0answers
92 views

Complexity class for probabilistic approximation algorithms with bounded error

What's the name of a complexity class of optimization problems that have "bounded error probabilistic approximation algorithms"? Bounded error probabilistic version of APX (as BPP is bounded error ...
0
votes
1answer
87 views

Using approximations to optimization problems for threshold problems

Many problems in computer science come in two flavors: Optimization problem: "Find an object with the largest size". Threshold problem: "Given $n$, find an object with a size of at least $n$, or ...
4
votes
1answer
157 views

NP-complete decision problems - how close can we come to a solution?

After we prove that a certain optimization problem is NP-hard, the natural next step is to look for a polynomial algorithm that comes close to the optimal solution - preferrably with a constant ...
1
vote
0answers
41 views

Quadratic programming problem involving permutation matrices

Does anyone know a good algorithm for quickly finding an approximate solution to the following problem? Given two square matrices $A$ and $B$, minimize $\| P A P^\top - B \|$ over all permutation ...
6
votes
1answer
90 views

Approximation algorithms for NP-complete problems

Given two NP NP-hard functional problems, A and B, one can find a reduction of A to B. Is it possible to find a reduction that would honour approximations? That is, if you have an approximation ...
1
vote
1answer
151 views

Finding an instance of an n-element set cover

Below is a homework problem where we have been asked to alter a greedy algorithm to return n element instance of a set problem. The original algorithm is also below. I was thinking that I could alter ...
1
vote
1answer
169 views

Traveling Salesman's Tour Approx Algorithm: is this really a Hamiltonian Path?

I'm given this problem: Consider the following closest-point heuristic for building an approximate traveling-salesman tour. Begin with a trivial cycle consisting of a single arbitrarily chosen ...
1
vote
0answers
161 views

Single machine job scheduling (Greedy heuristic)

Here is a variation of a job-scheduling Problem. Let $J = \{j_1,...j_n\}$ be a set of Jobs for $1 \leq i \leq n$. Given Job length $|j_i|\in \mathbb{N}$, deadline $f_i \in \mathbb{N}$, profit $p_i \ge ...
4
votes
2answers
133 views

NP-hardness and FPTAS

I have a problem in understanding how to prove the following question. Let $Q = \langle\max,f,L\rangle$ be an NPO-Problem, where $f$ only supports integers. Define $$L_Q^* =\{(x_0,1^k) : \exists x . ...
1
vote
1answer
364 views

Approximated TSP: weight of minimum spanning tree less than cost of the optimal tour?

In the chapter, Approximation Algorithms of Introduction to Algorithm, 3rd Edition, for the approximation problem Travelling Salesman Problem, the author proposes a approximation method that first ...
2
votes
1answer
211 views

without triangle inequality, finding good approximate tours for TSP in polynomial time is impossible unless P=NP?

In the text book, Introduction to Algorithm, 3rd Edition. In the chapter, Approximation Algorithms and for the problem Travelling Salesman Problem, the author says: I am wondering how triangle ...
-1
votes
1answer
41 views

Is a non-perfect improvement and optimisation?

In real word problems, the influence of multiple not perfectly known factors results in using heuristics instead of mathemacial solutions that calculates a perfect value from only precisly defined ...