Questions tagged [approximation]

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

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40 votes
3 answers
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Decision problems vs "real" problems that aren't yes-or-no

I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. ...
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10 votes
1 answer
178 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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13 votes
1 answer
1k views

Why are all problems in FPTAS also in FPT?

According to the Wikipedia article on polynomial-time approximation schemes: All problems in FPTAS are fixed-parameter tractable. This result surprises me - these classes seem to be totally ...
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18 votes
2 answers
12k views

PTAS definition vs. FPTAS

From what I read in the ...
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11 votes
2 answers
4k 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: cis.upenn....
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13 votes
1 answer
17k 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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-1 votes
2 answers
1k views

Vertex cover of bipartite graph

A vertex cover is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A minimum vertex cover is a vertex cover with minimal cardinality. From codeforces, ...
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31 votes
7 answers
39k 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 ...
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  • 433
8 votes
3 answers
654 views

What is the name of this logistic variant of TSP?

I have a logistic problem that can be seen as a variant of $\text{TSP}$. It is so natural, I'm sure it has been studied in Operations research or something similar. Here's one way of looking at the ...
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5 votes
1 answer
744 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 ...
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5 votes
0 answers
656 views

2 Dimensional Subset Sum: looking for information

I do not know if this problems exists with a different name, if it is, I could not find it. The problem is this: Given a set $S$ of $n$ points in $\mathbb{Z}^2$, is there a subset $A\subset S$ ...
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  • 111
4 votes
2 answers
3k 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 ...
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1 vote
1 answer
630 views

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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23 votes
3 answers
2k views

Approximating the Kolmogorov complexity

I've studied something about the Kolmogorov Complexity, read some articles and books from Vitanyi and Li and used the concept of Normalized Compression Distance to verify the stilometry of authors (...
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10 votes
3 answers
10k views

Difference between heuristic and approximation algorithm?

i have a problem regarding the following situation. I have two arrays of numbers like this: ...
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  • 337
9 votes
2 answers
2k views

Looping and branching with Algorithmic Differentiation

Algorithmic (aka Automatic) Differentiation is a wonderful technique for numerical computation of derivatives. I understand how it relates to the fact that we know how to deal with every elementary ...
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  • 205
3 votes
1 answer
6k views

Correctness proof: 2-approximation of greedy matching-algorithm

Input: number of edges and vertices, and array of all edges in graph. Output: array of edges that construct a matching, so that: $$\frac{\text{the number of edges in this matching}}{\text{the number ...
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  • 33
2 votes
1 answer
2k views

Bottleneck TSP with MST

There is a problem I don't know the answer too. The 3 approximation for the bottleneck TSP that involves first getting the MST. I have not been able to come up with the right "shortcut" method so far. ...
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  • 39
6 votes
2 answers
970 views

Example for a non-trivial PCP verifier for an NP-complete problem

During my involvement in a course on dealing with NP-hard problems I have encountered the PCP theorem, stating $\qquad\displaystyle \mathsf{NP} = \mathsf{PCP}(\log n, 1)$. I understand the ...
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  • 71k
5 votes
1 answer
878 views

Approximation algorithm for TSP variant, fixed start and end anywhere but starting point + multiple visits at each vertex ALLOWED

NOTE: Due to the fact that the trip does not end at the same place it started and also the fact that every point can be visited more than once as long as I still visit all of them, this is not really ...
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  • 53
5 votes
2 answers
440 views

Approximation ratio when optimal solution is $0$

This might be a basic technicality but I'd like to make sure how to handle it. The question is: how do we measure an algorithm's approximation (multiplicative) factor on instances with optimal value $...
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5 votes
1 answer
363 views

Minimal Steiner Tree in unweighted directed graph

I have an unweighted directed graph $(V, E)$ and a subset $T \subseteq V$ of these vertices. I want to find the minimum tree $(V',E')$ that contains all these $T$ vertices (minimize in number of nodes ...
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3 votes
2 answers
401 views

Minimising sum of consecutive points distances Manhattan metric

I have two sets $X$ and $Y$ of 2-dimensional points. The points are floating point numbers. The objective is to sort them in such way that sum of differences in distances of consecutive sorted points ...
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  • 9,325
2 votes
1 answer
218 views

Why the Goemans-Williamson's MAX-CUT algorithm relax the variables to vectors of $n-$dimension on unit sphere?

Why not to some constant like 3 or 4 dimension? I suspect that it is because Cholesky Decompostion will work only for $n \times n$ matrix $B$ where $B^TB = P$ where $P$ is a semidefinite matrix. Is it ...
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1 vote
1 answer
125 views

Seeking Efficient Approximation Algorithm for Adaptation of TSP

Consider the following adaptation of the traveling salesman problem: Given a complete, undirected graph $G$ with nonnegative edge weights, color each vertex either red or blue. Find the shortest ...
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  • 11
4 votes
1 answer
189 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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  • 329
4 votes
1 answer
793 views

Best data structure for high dimensional nearest neighbor search

I'm actually working on high dimensional data (~50.000-100.000 features) and nearest neighbors search must be performed on it. I know that KD-Trees has poor performance as dimensions grows, and also I'...
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  • 141
3 votes
5 answers
917 views

What does big O mean as a term of an approximation ratio?

I'm trying to understand the approximation ratio for the Kenyon-Remila algorithm for the 2D cutting stock problem. The ratio in question is $(1 + \varepsilon) \text{Opt}(L) + O(1/\varepsilon^2)$. ...
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3 votes
1 answer
902 views

Big-O / $\tilde{O}$ -notation with multiple variables when function is decreasing in one of its arguments

Say we have an algorithm that takes an input a triple ($X$, $A$, $\epsilon$), where $X$ is a sequence of $n$ bytes, of which the algorithm might query only a subset, and $A$ and $\epsilon$ are ...
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2 votes
1 answer
1k views

How to correctly define the ratio of an approximation algorithm?

For a maximization problem $P$, I know that an $\gamma$-approximation algorithm for $P$ produces a solution $S$ that is $|OPT|\ge |S| \ge \gamma\cdot|OPT|$ for $\gamma <1$ and $OPT$ the optimal ...
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  • 305
1 vote
1 answer
50 views

Communication complexity of equality gap problem

I'm interested to know what is the biggest known $0\le \epsilon\le 1$ such that the $gap-EQUALITY$ problem that is defined by: $$f_\text{GEQ}(x,y)=\cases{1&$x=y$\\0 & $x$ and $y$ differ in at ...
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  • 10.8k
1 vote
1 answer
82 views

About showing algorithmic gap instance for the Goemans-Williamson SDP

Using usual notation we have, $SDP(G) \geq OPT(G) \geq Alg_{GW}(G) \geq \alpha_{GW} SDP(G) \geq \alpha_{GW} OPT(G)$ where we mean, $SDP(G)$ = The maximum value that the SDP finds of the objective ...
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-1 votes
1 answer
72 views

acyclic and disjoint union

I would like to find a prove of (a) so that the two E are acyclic and disjoint union and I dont unterstand b Could someone shed light on this problem, preferably spiced with some intuition? Thanks, ...
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