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Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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1answer
70 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 $k$ ...
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1answer
420 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 $...
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1answer
64 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 ...
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53 views

Rules of thumb for universal approximation

A neural network can function as an universal approximator for a function $f$, according to the universal approximation theorem (original paper). In this answer, I read that "As a rule of thumb, each ...
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1answer
123 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, $k$...
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1answer
99 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 $U\subseteq V$...
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0answers
155 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 ...
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2answers
416 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 ...
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128 views

How to partition a graph while minimizing the count of intra-edges?

Suppose we are given an undirected graph $G(V,E)$. The objective is to partition the graph vertices $V$ into $n$ partitions $P_1, P_2, ..P_n$ such that following criteria is satisfied: $|P_i| = |V|/n$...
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54 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, ...
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56 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 ...
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83 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:...
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100 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 ...
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28 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
81 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 $\...
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0answers
58 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 ...
3
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1answer
171 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) ...
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0answers
214 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 ...
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1answer
679 views

Is the current best approximation ratio for Vertex Cover problem also a lower bound?

In textbook "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. in pp.1110-1111, they argue that the vertex-cover problem is a 2-approximation algorithm and it is lower bound so we ...
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0answers
195 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 ...
4
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1answer
212 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 ...
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3answers
161 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 ...
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0answers
168 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 ...
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0answers
122 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 ...
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152 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 ...
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0answers
103 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 ...
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0answers
128 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 ...
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0answers
155 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
187 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
193 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 ...
2
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1answer
83 views

Density of multiples

I have an infinite collection of positive integers $n_1,n_2,n_3,\ldots$ and I would like to find the density of the numbers divisible by one or more of these.* If the density does not exist, the ...
6
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1answer
138 views

Brute force search algorithm for semidefinite programming (representation of spectrahedron)

I was wondering if there exists a brute force search algorithm for semidefinite programming problems. Specifically, can we find finite number of points in the positive semidefinite cone such that for ...
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1answer
118 views

Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
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1answer
177 views

Online/approximate weighted and capacitated bipartite matching

I wish to take a look at online/approximate weighted and capacitated bipartite matching problem. Consider $G=\{L\cup R, E\}$, $|L|=n_1$, $|R|=n_2$, $|E|=m$ and $E\subseteq L\times R$. For each $r_i\...
6
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1answer
306 views

Ordering of a DAG minimizing some definition of cost

Consider a DAG $(V,A)$ with a topological ordering $(v_1,v_2,\ldots,v_n)$. I define the cost of this ordering as the maximum over all $1\leq i\leq n$ of $|\{j\leq i \mid \exists k>i: (v_j,v_k)\in A\...
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2answers
187 views

Solution/Hardness of the following (integer) budgeted problem?

I have no idea how to solve the following INTEGER problem or prove its hardness. Thanks for any help/comment/open discussion! Assume there are $N$ startups. For each startup $i$, you can invest $x_i\...
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0answers
104 views

algorithms for a large submatrix / general factor / quasi-biclique problem?

Given a sparse 0/1 matrix $X$, too large to fit in memory, with $m$ rows and $n$ columns, I'm looking for an algorithm for finding a submatrix (when one exists) with maximum number of rows such that ...
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0answers
306 views

Efficient algorithms for counting $k$-clique subgraphs

Given a graph $G$ with $n >> k$ vertices, what are the fastest algorithms known to count the number of induced subgraphs in $G$ that are $k$-cliques? Are there algorithms that can do better ...
3
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1answer
205 views

How can we bound the optimal solution of the dual bin packing when we solve the knapsack problem for each bin?

I have these two problems: Problem 1 (Dual bin packing problem) Instance: A set of $n$ items where each item $i$ has weight $w_i$. A set of $k$ bins where each bin has capacity $W$. Question: Find ...
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0answers
95 views

Percolation probabilities

I have a finite undirected graph with a probability $p_e$ given for each edge $e$. This gives a random graph by removing each edge e with probability $1-p_e$ independently of the others. I'm ...
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121 views

Weighted $l_1$ distance

So there are many well known algorithms for approximate nearest neighbor on the $\ell_1$ distance. My question is, what about the weighted version of the problem (where the weights are specified along ...
2
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1answer
592 views

Max-sum graph-partition for maximizing intra-edge weights?

I would like to know if the following problem has already been studied, and if so how is it called. In particular I'm interested in approximability results. Input: A graph G with negative or non-...
3
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1answer
454 views

Partial cover approximation

We have a set of elements $E=\{e_1, e_2, \ldots, e_m\}$, and $n$ subsets of $E$: $S_1, S_2, \ldots, S_n$ The union of those subsets is $E$, and each subset $S_i$ has a non-negative weight $w_i$. The ...
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0answers
151 views

Fast Approximation Algorithms for Covering Design

The covering design problem is as follows: We are given a universe $\mathcal{U}$ of size $n$. By $C(n,k,l)$ we denote the smallest cardinality of any set system $\mathcal{A} \subset 2^\mathcal{U}$ ...
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0answers
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About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
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1answer
144 views

Does k-PATH admit a constant approximation?

In the $k$-PATH problem, we receive as input a graph $G$ and an integer $k$. The goal is to decide whether there exists a simple path of length $k$ in $G$. A $\alpha$-approximation for $k$-PATH is an ...
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0answers
38 views

Stochastic optimization with erroneous oracles

I am interested in a class of optimization problems of which we know that the input variable is first subjected to noise $\xi$ before entering the data-producing process $f$. I write the objective in ...
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1answer
642 views

Maximizing a monotone supermodular function s.t. cardinality

I've tried to comb the literature and seen a lot of references to results that almost but don't quite seem to address this. Question: Is it known to be true or is there a hardness result ...
3
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2answers
430 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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0answers
241 views

Maximizing the number of selected edges with opposing requirements

Consider the following problem: Input: a complete bipartite graph $G$ with its edges colored either white or black, a number $k$. Output: a subset of vertices $W$ of size $k$ which maximizes the ...