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Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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Which is the approximation class of Minimum 0-1 integer programming if only non-negative integers are allowed?

I'm wondering which is the approximation class of Minimum 0-1 Integer Programming if only non-negative integers are used. That is: minimize $c^T \cdot x$, with $x\in \{0,1\}^n$ and $c\in (\mathbb{Z}^...
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74 views

Distributing bags of apples equally

Assume you have $N$ bags of apples that you want to equally distribute to $K$ people. Each bag contains $n_i$ apples and you are not allowed to open and divide the bags; you must distribute the bags ...
3
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1answer
244 views

Approximation algorithms for the maximum $2$-independence set problem

I am interested in approximating the maximum $2$-independent set problem in arbitrary graphs. In a graph $G$ a set $I$ of vertices is called $2$-independent if the distance between any two distinct ...
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204 views

Linear Programing with Rounding for the Fire Station Problem

Consider the following fire station problem: The input is a positive integer k and a complete undirected graph $G = (V,E)$ with distances on the edges. The distances form a metric: $d(v, v) = 0$, $d(...
3
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1answer
191 views

The complexity of decomposing a bi-stochastic matrix

A bistochastic matrix $A$ is a matrix with positive entries in which each row/column sums to $1$. By the Birkhoff von-Neumann theorem $A$ is a convex combination of permutation matrices. Further, by ...
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33 views

On approx-preserving P- and A-reducibilities

Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$, in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time ...
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114 views

Maximize number of bins and minimize cost of elements chosen from a set

I am considering the following problem: there is a set of elements $S$ where each element is assigned to a bin $B$. The bins are disjoint and their union is $S$. There is also a cost function ...
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1answer
79 views

Approximations for the Stable Fixtures Problem

I have a set of N items, each with a subset of those items they can be paired with; each pair has a weight. I'd like to choose pairs to maximize the total weight, subject to each item being in at ...
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1answer
303 views

Trying to understand a paper on ksvd algorithm (dictionary learning) by Elad, et al

Trying to understand a paper titled KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation by M.Elad, et al; my take of section IV.C. detailed description of KSVD, is ...
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2answers
237 views

The complexity of finding a Borsuk-Ulam point

The Borsuk-Ulam theorem says that for every continuous odd function $g$ from an n-sphere into Euclidean n-space, there is a point $x_0$ such that $g(x_0)=0$. Simmons and Su (2002) describe a method ...
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1answer
466 views

Constrained version of vertex cover in a bipartite graph

Let $G(V_1, V_2, E)$ be a bipartite graph such that degree of all the vertices in $V_1$ is bounded by some constant (say) $d$. Now, for given two positive integer $l$ and $k$, we wish to decide if ...
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Which matrix of Q values is being used here?

This question refers to this paper: Using Free Energies to Represent Q-values in a Multiagent Reinforcement Learning Task In section 2.1, equations (5) and (6), I am wondering which Q values are ...
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1answer
86 views

Approximate matching in table of integer vectors

Disclaimer: This is my first question on cstheory.stackexchange.com so please be forgiving. I have a list of M (M is big, more than 1 million elements) vectors of integers. Each vector can contain 0-...
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Is there any exsiting research on this kind of “sorting with constraint” problem?

I have been interested in this kind of "sorting with constraint" problem: Given $n$ items $\{S_1 ,S_2 ,...S_n\}$ with corresponding weight $w_i ,i=1,2,...,n$, we want to sort these $n$ items (i.e. ...
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76 views

“conservative approximate Set Cover”?

We are given a lattice graph $L$, embedded on the plane, and a certain "shape" (a connected, acyclic subgraph $S$ of $L$). The task is to approximately cover $L$ with translated, rotated and flipped ...
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2answers
255 views

What are some of the most ingenious linear programs developed for tackling hard combinatorial problems?

What are some known ingenious linear programs that have been developed for tackling hard combinatorial optimization problems, especially any linear programs which had helped in getting good ...
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114 views

An algorithm to compute sine and cosine within $n$-digit precision

I know CORDIC, but is there a better algorithm to compute $\sin x$ and $\cos x$ within specified $n$-digit accuracy?
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Statistical Algorithms vs Convex Relaxations - Planted Clique

I am trying to understand exactly what the lower bounds for the query complexity of statistical algorithms imply for convex relaxations for the planted clique problem ? A recent paper by Feldman, ...
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130 views

Difficult On Average Cases for 3MaxSAT and 3SAT Approximation Algorithm [closed]

1.Its known that a polynomial time approximation algorithm that satisfies 3MaxSAT in 7/8+e clauses implies P=NP. 2.Its also experimentally known that 3SAT has the most difficult known cases when the ...
2
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2answers
280 views

Coreset and VC dimension

I am trying to understand the notion of $\epsilon$-coreset and its relation with sampling bounds of a range space having a finite VC-dimension. Although both of them give an $\epsilon$-approximation ...
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301 views

How good is greedy in average?

Given a family ${\cal F}\subset 2^E$ of (feasible solutions), the maximization problem on ${\cal F}$ is, for every weighting $x:E\to \{0,1,\ldots\}$ of ground elements, to compute the maximum weight ...
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What is known about this TSP variant?

This question was previously posted to Computer Science Stack Exchange here. Imagine you're a very successful travelling salesman with clients all over the country. To speed up shipping, you've ...
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1answer
254 views

Are there any learning algorithms with any provable guarantees for manifold learning or manifold regularization?

First of all, I want to make clear that my question is about algorithms. I'd like to know if there are any algorithms with provable guarantees in the context of manifold learning (or manifold ...
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1answer
218 views

Approximating the clique size of the graph

Let $G=(V,E)$ be a graph. For a given $\rho \leq |V|$ and $\epsilon$ with $(0<\epsilon<1)$, is there any sublinear query algorithm known/possible to decide if the graph has a clique of size $\...
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117 views

3D Bin Packing with one bin with infinite/unknown size

Hi I'm looking for a variation of the Orthogonal 3D-BinPacking algorithm with only one bin of unknown size. I have a set $S$ of $n$ cuboids items $i_j$ with $j=1...n$. The dimensions of the items are ...
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1answer
138 views

What is the intuition behind “hardness of approximation”? [closed]

I am reading a paper about graph matching problem. Which is, to some extent, an optimization version of the graph isomorphism problem. To my surprise, some closely related NP-hard problems are quite ...
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132 views

Exactly solvable but non-trivial integrality gap

Are there interesting polynomial time solvable problems that we know of for which the natural convex relaxation has a non-trivial integrality gap? Note: Maximum matching doesn't qualify because I ...
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1answer
393 views

What are multiple rounds of SOS/Lasserre hierarchy?

Is that the same as saying the one will try to generate a higher-degree "pseudo expectation functional" by solving a SOS-program ? Or is there a difference between the two things? Or to take a ...
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1answer
422 views

The Goemans-Williamson algorithm in the $SOS$ framework

If there is a variable $x_i$ for every vertex $i$ of a $d$-regular graph $G$ then assigning $x_i = \pm 1$ gives a cut, say $(S,\bar{S})$, of the graph. We can then see that, $\langle x,L x\rangle$, ...
1
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1answer
220 views

Evaluating the expected value of negatively correlated random variables

A polynomial random process satisfying the following properties converts a fractional point $(x_1, x_2, \ldots, x_n) \in \mathcal{P}$, $(x_i \in [0,1])$ to a random integer point $(X_1, X_2, \ldots, ...
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Is MAX-SAT SETH (like) hard?

If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy. There ...
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1answer
409 views

Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning?

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). I am particularly ...
5
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1answer
208 views

Set cover in which some pairs of sets are forbidden

I'm trying to find an approximation algorithm for a variant of the weighted set cover problem. However, this variation doesn't seem to let me apply the traditional set cover arguments for an ...
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0answers
110 views

Multiple knapsack fpt?

It was an open question whether multiple knapsack is fpt wrt standard parameter. Since at SODA 2009 Jansen has presented an EPTAS for multiple knapsack and an EPTAS implies the existense of an fpt ...
3
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1answer
2k views

Algorithms for computing the minimal vertex separator of a graph

Background: Let $u,v$ be two vertices of an undirected graph $G=(V,E)$. A vertex set $S\subseteq V$ is a $u,v$-separator if $u$ and $v$ belong to different connected components of $G−S$. If no proper ...
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1answer
265 views

NP-complete problems with optimal approximation in poly-time

I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not $1+\...
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103 views

$\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ problems

All the problems which are either $\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ cannot admit a $\mathsf{PTAS}$, unless $P=NP$. I would like to know whether $\mathsf{MaxSNP}$ ...
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212 views

Steiner Tree and minimum spanning tree

If I must connect: $$2^k$$ terminals in a Steiner Tree choosen randomly and connect them with the cheapest component; "loss - contracting algorithm" is a good way? Or is an "Iterative Randomized ...
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88 views

Approximation Algorithm for TSP-like problem

Suppose we are given a graph with distances for each of the edges and merit for each of the nodes. What are the best (approximation) algorithms for computing the the most meritorious simple path with ...
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228 views

Recent insights on algorithms for 1D bin packing

This is just a general question on recent algorithms for the 1D bin packing problem. I just want to collect some information on this issue, so I’m grateful for any information. Especially heuristics ...
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1answer
990 views

Modifying Hopcroft-Karp algorithm to get approximate bipartite matching

I am trying to find an algorithm to find an $\epsilon$-approximate maximum matching $M_{\epsilon}$ in a bipartite graph in $O(m/\epsilon)$. The partite groups are of equal size, they are $A$ and $B$. ...
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112 views

The distribution on the solution space induced by randomized rounding

Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if $maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut $S$ of fractional size at least $1-\sqrt{\epsilon}$...
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79 views

k closest points that belong to a set

This is a question from theory community, but I came across this issue in a practical problem. So just have this in mind. I have a set of real vectors: $$ S = \lbrace v_1, \dots, v_n \rbrace $$ $...
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99 views

Approximating a max-cut's intersection with other cuts

For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set $S$ and its complement. A max cut is one with at least as many edges as any ...
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Advances towards proving the Held-Karp conjecture for TSP

I've only began my research into the Held-Karp conjecture and I was wondering about recent progress in proving the conjecture. The Held-Karp relaxation is conjectured to have an integrality gap of $\...
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1answer
272 views

Approximation algorithms for min vector subset-sum over GF(2)

In this question vzn asked about the following problem, which I'll call Vector-Subset-Sum. Given a set of vectors $v_i$ over GF(2) and a target vector $y$, is there a subset of the $v_i$ summing to ...
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Maximizing a submodular function with restricted values

Maximizing a general monotone submodular function $f$ under the constraint that $|S|\leq k$, can be approximated to $(1-1/e)$. I am wondering if a better approximation algorithm exists if the ...
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612 views

Practically Good Algorithms of a Very Low Computational Complexity Class

I am looking for one (or more) examples of a parametric class of algorithms $P_t$ for approximately solving a class $\cal A$ of algorithmic questions with the following properties: 1) Solving the ...
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2answers
699 views

What is this variant of set cover problem known as?

Input is a universe $U$ and a family of subsets of $U$, say, ${\cal F} \subseteq 2^U$. We assume that the subsets in ${\cal F}$ can cover $U$, i.e., $\bigcup_{E\in {\cal F}}E=U$. An incremental ...
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2answers
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Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup B=...