Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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3
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0answers
44 views

Approximating a monotone submodular function using a concrete coverage function

Is it possible to approximate a monotone submodular function using a concrete coverage function, i.e. Given a ground set $U$ and a monotone submodular function, $f:2^U\to \mathbb{R}$, the goal is to ...
2
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1answer
214 views

Monotone supermodular function minimization under partition matroid constraints

Is there a known approximation algorithm for the problem of minimizing a monotone (non increasing) supermodular function under partition matroid constraints ?
4
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0answers
111 views

Learning hidden variable distribution

Consider a set of $k$ continuous variables. Each variable $x_k$ is associated with a hidden distribution from which its value is sampled independently of other variables. I am given a set of ...
3
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0answers
124 views

Stacks serving interval storage requests

An interval storage request is represented by a tuple $(s,t,v)$ satisfying $s<t$, meaning that the value $v$ needs to be stored from time $s$ to time $t$. A stack serves the request $(s,t,v)$ in ...
2
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0answers
96 views

Unbalanced connected partition

Let $G = (V, E)$ be a connected graph with (possibly negative) vertex weights $w(v)\in\mathbb{Z}$. We want to partition the vertices into two parts such that the induced graphs $G'$ and $G''$ are ...
14
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2answers
594 views

Proof assistant usage in complexity theory research?

Considering the topics covered at a conference like STOC, are any algorithm or complexity researchers actively using COQ or Isabelle? If so, how are they using it in their research? I assume most ...
0
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2answers
201 views

Geometric max cover

Consider $n$ points and a distance function $d$ that satisfies the triangle inequality. We are also given a number $r$. Each point $p$ defines a set $B_p$ (or a ball) that covers all other points ...
1
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1answer
624 views

Common terminology used for lower/upper bounds

Suppose you have developed an upper bound on the number of vertices of a particular graph. This bound is the best possible bound that can be found for any given instance. What do you call such a bound?...
3
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0answers
114 views

On approximating problems in $\#P$

We know that for every counting problem $\#A$ in $\#P$, there is a probabilistic algorithm $\mathcal C$ that on input $x$, computes with high probability a value $v$ such that $$(1 − ε)\#A(x) ≤ v ≤ (1 ...
10
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1answer
402 views

Find an approximate argmax using only approximate max queries

Consider the following problem. There are $n$ unknown values $v_1, \cdots, v_n \in \mathbb{R}$. The task is to find the index of the largest one using only queries of the following form. A query ...
2
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0answers
61 views

\alpha-path on Euclidean graphs

Consider the following problem: Suppose we are given a G=(V, E) Euclidean Graph in the plane and a real $\alpha > 0$. For simplicity assume, there exists only one path whose summation of weights ...
6
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0answers
156 views

Lower bound for Yao's algorithm on general addition chains?

An addition chain of size $n$ for given integers $n_1,n_2\dots ,n_p$ is a sequence of integers $k_1,k_2\dots ,k_n$ such that $k_1=1$, for all $i$ $(2\le i\le m)$ we have $k_i=k_j+k_m$ for some $1\le ...
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0answers
103 views

Hausdorff Distance and Convex Hull

Given two sets of points A and B, both in $R^d$, is there a relation between the convex hulls of A and B, i.e. conv(A) and conv(B), w.r.t. the Hausdorff distance between A and B? In other words, does ...
2
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0answers
63 views

On the impossibility of representing/approximating subadditive function using additive functions

I am trying to understand whether a monotone and subadditive function $f(S), S \subseteq 2^{[n]}$ and $ f : 2^{[n]} \rightarrow R_{\geq0}$ can be represented using an additive function $\hat{f}(S) = \...
1
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1answer
121 views

Practical/heuristic algorithm for multi set-cover

Consider a universe $N$ containing $n$ elements, and a collection of sets $\mathcal{C}$, over $N$. The $k$-multiset multicover (MSMC) problem is to cover all elements of the universe $N$ at least $k$ ...
3
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1answer
140 views

Is it possible to approximate Maximum Independent Set in $O(2^k\text{poly}(n))$ time?

We know that MIS is hard to approximate within a $n^{1-\epsilon}$ factor in polynomial time and that it is $W[1]$-hard and thus unlikely to admit a $f(k)\text{poly}(n)$ time exact algorithm. (here, $k$...
1
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1answer
618 views

Clique cover problem

Consider the following graph problem. We are given a graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. For each vertex $...
0
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1answer
71 views

Approximating max degree $3$ perfect matching count?

We do not have a deterministic constant factor approximation scheme for general $n\times n$ $0/1$ permanent. What is the best factor in deterministic approximation schemes if we only care counting ...
5
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0answers
221 views

Why is HyperLogLog (near-)optimal?

The original HyperLogLog paper claims that this probabilistic counting algorithm is "near-optimal". The relevant section of the paper reads: Clearly, maintaining $\epsilon$-approximate counts till ...
1
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1answer
115 views

hardness of constant approximation of largest matching set

We say that $H$ is a matching graph if it contains $2n$ vertices and only $n$ vertex-disjoint edges, i.e. $H$ only contains those $n$ edges and no more. Given a graph $G=(V,E)$ a subset $U\subseteq V$...
2
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0answers
175 views

Approximating the Radius of a (Dense) Graph

For a (dense) graph, computing its radius is as hard as computer "All Pairs Shortest Paths" (APSP) [1]. So we can focus on approximating the radius. A $(1+\epsilon)$-approximating of APSP for a ...
1
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0answers
69 views

Sparse-cut approximation for well connected graphs

Theorem: For any graph $G=(V,E)$, there is a polynomial time algorithm that finds a cut $S \subseteq V$ with conductance at most $\sqrt{2\big(1 − \lambda(G)\big)}$. If I understand UGC correctly, ...
2
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0answers
58 views

Studying the hardness of polynomial-time approximability using the concept of stability of approximation

In the Conclusion section, the author of this paper "Stability of Approximation Algorithms for Hard Optimization Problems" by Juraj Hromkovič, 1999 claims that Using the notion of stability one can ...
7
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0answers
112 views

Deterministic approximation algorithms for treewidth

As far as I understand, the factor $O(\sqrt{\log OPT})$ approximation algorithm for treewidth of Feige, Hajiaghayi, and Lee is randomized, and no deterministic approximation algorithm with this factor ...
2
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0answers
104 views

Alternative Set Cover Algorithm With Doubling

I remember that I saw once an alternative to the greedy set cover algorithm that works as follows: Assign weight 1 to every element in the universe. Repeat steps 2 and 3 until the universe is covered:...
2
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2answers
340 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 ...
4
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1answer
246 views

Looking for approximation class between NPO and Exp-APX

I'm trying to identify the approximation hardness of some maximization problem A. In problem A, finding a solution whose quality is 0 (i.e. such that the value returned by the objetive function is 0) ...
2
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0answers
34 views

Sparse coding and matching pursuit algorithms

Is it true that all known sparse coding algorithms which work efficiently in practice don't have convergence proofs and always use an intermediate step of a matching/subspace pursuit algorithm on the ...
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0answers
84 views

Is it sufficient to only check on the vertices? Greedy algorithm

Suppose that I have a downwards closed Polyhedron and a vector $\omega$ and a greedy algorithm that goes as follows: Given an initial $x_0$, order the indices of $\omega$. Then for each $i$, solve $\...
1
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0answers
61 views

What is known about data structures for encoding a set while considering approximate Rank queries?

Consider a universe $\mathcal U\triangleq \{1,2,\ldots n\}$, and assume that we are given a set $S\subseteq \mathcal U$. There are many data structures that allow storing $S$ while answering Rank ...
11
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2answers
4k views

Any fast algorithm for minimum cost feedback arc set problem?

In a directed graph, $G=(V,E)$, $F\subset E$, if $G\setminus F$ is a DAG(directed acyclic graph), $F$ is called a feedback arc set. If each edge is associated with a weight $w$, the minimum cost ...
7
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1answer
315 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 ...
5
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0answers
202 views

On permanent of $\{\pm1,0\}$ matrices

Consider the problem of computing the permanent $Per(M)$ of a matrix $M\in\{0,-1,1\}^{n\times n}$ such that the result is bounded in absolute value, $|Per(M)|<B$ where $B$ is part of input. Is ...
1
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0answers
873 views

Finding minimum weight $k$ cliques in a complete graph

For an undirected weighted complete graph $G = (V, E)$. Assuming the edge weight indicates the similarity between different nodes, the smaller $w_{ij}$ is, it means $i$ and $j$ are more similar ...
3
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1answer
226 views

Compression algorithms for low-complexity strings?

Let $s$ denote a string over a finite alphabet, $n_s = |s|$ be the length of $s$, and $n_s^{*}$ denote the minimum description size of $s$ under a given computational model (TM, CFG, etc.). Are there ...
3
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3answers
172 views

Approaches for Theoretical Analysis of Estimates of Probability Distributions

Consider that you have a probability distribution p of some quantity X and you have obtained (via some algorithm) an estimate q of p according to some definition of closeness. Are there methods/papers ...
1
vote
1answer
1k views

Difference between Multiple Knapsack problem and Multidimensional Knapsack Problem

What is the difference between Multiple Knapsack problem and Multidimensional Knapsack Problem? (http://en.wikipedia.org/wiki/List_of_knapsack_problems#Multiple_constraints) According to the ...
4
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0answers
332 views

What does “no integrality gap” imply?

I'm currently working on a linear time heuristic for the rectangle decomposition of a binary matrix. This problem has a polynomial time solution, which in our case is too slow for large-scale ...
45
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8answers
6k views

The importance of Integrality Gap

I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real ...
5
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1answer
742 views

Is there an approximation algorithm for MAX k DOUBLE SET COVER?

Given a set system $(X,\mathcal S)$, let us say that a subset $\mathcal C\subseteq \mathcal S$ doubly covers a vertex $x$ in $X$ if $x$ is contained in at least two sets of $\mathcal C$. Let us define ...
1
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0answers
191 views

Maximize the weight of MST + sum of vertex weights

I am considering a problem where the goal is to choose a subset of size $k$ of the vertices in a graph, such that the weight of their minimum spanning tree + the sum of their vertex weights is ...
3
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0answers
126 views

Given a matrix $A$ maximize the number of positive elements in $Ax$ under specific constraints for $x$

Let $A = [a_{ij}]$ be a symmetric matrix with nonnegative values and $k << n/2$ a given constant. We want to rearrange the columns of the matrix such that the number of rows with the following ...
6
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0answers
180 views

Positive cut algorithm on bipartite graphs with negative weights

Let $G=(V,E,w)$ be a bipartite graph with weight function $w:E→\{-1,1\}$. Is there an efficient (polynomial) algorithm for finding some positive (not necessarily maximum) cut of $G$, if one exists? If ...
35
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3answers
4k views

Max-cut with negative weight edges

Let $G = (V, E, w)$ be a graph with weight function $w:E\rightarrow \mathbb{R}$. The max-cut problem is to find: $$\arg\max_{S \subset V} \sum_{(u,v) \in E : u \in S, v \not \in S}w(u,v)$$ If the ...
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0answers
118 views

Truthful posted-price mechanism with optimal efficiency (social welfare)

I am interested in mechanism design. In the paper On Profit-Maximizing Envy-free Pricing, SODA, 2005, the authors provided a truthful competitive posted-price mechanism with $4\log h$ guarantee of ...
3
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0answers
143 views

Maximize number of edges covered by an independent set of vertices

Smallest vertex cover which is also an independent set asks about finding an independent set that covers all edges. This problem is known as the independent vertex cover problem and is equivalent to ...
1
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0answers
159 views

Algorithms to approximate a Riemann integrable function by a piecewise constant function

We know that any general function that is Riemann integrable can be approximated to arbitrary precision by piecewise constant functions. So in this regard, I was looking for references on algorithms (...
6
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2answers
267 views

Finding smallest context free grammar that generates a set of sets

Are there any results known about the size of smallest context free grammar that generates a set of sets? That is, I am given an alphabet $\Sigma$ as well as a set $S \subseteq \mathbb{P}(\Sigma)$ ...
3
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0answers
308 views

Approximating the VM packing problem

In the wikipedia article on bin-packing it is stated that A variant of bin packing that occurs in practice is when items can share space when packed into a bin. Specifically, a set of items could ...
3
votes
2answers
488 views

Quantum complexity of maximum inner product search

Given two matrices $X \in \mathbb{R}^{m \times k}$, $Y \in \mathbb{R}^{n \times k}$, maximum inner product search (MIPS) asks for the largest $l$ entries of $X Y^T$. Typically $k \ll m, n$ (many ...

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