Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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Are there good analogues to Sparsest Cut/Balanced cut for vertex separators instead of edges cuts?

Most problems about cutting graphs into roughly equal parts such as Sparsest cut, Graph partition, Balanced Cut, etc are based on minimizing the size of an edge cut. Even if all of those problems are ...
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Find the minimum cost spider joining a root to some leaves

A spider is a tree with at most one vertex of degree greater than 2. This vertex is called the head of the spider. I am interested in the following problem: We are given an undirected graph $G = (V,E)$...
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1 answer
101 views

Which is the most efficient of the two following approximation algorithms?

Let $\mathfrak{B}$ be a set, which will be called the set of bins. Suppose we have five maps \begin{align*} \mathrm{Value} &: \mathfrak{B} \to \mathbb{R} \\ \mathrm{Upper} &: \mathfrak{B}...
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The tree augmentation problem, but with hyperlinks

In the (Weighted) Tree Augmentation Problem, we are given a tree $T = (V,E)$ and a set of additional edges $L$ called links with non-negative costs. Each link $\ell = (u,v)$ covers the tree edges ...
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Approximate inclusion-Exclusion?

I am trying to understand or find literature on the following problem of approximate inclusion exclusion. Let $S:=\{A_i\}_{i=1}^{m}$ be a set of $m$ sets. Every intersection of $k$ elements in $S$ ...
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Is there FPT or XP algorithms known for Shortest Steiner cycle and $(a,b)$-Steiner path problem

Shortest Steiner cycle and $(a,b)$-Steiner path problem are generalizations of optimization versions of Hamiltonian cycle and Hamiltonian path problems. The Shortest Steiner cycle problem is defined ...
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131 views

The Complexity of Multi-Objective Optimization

Given a vector set $V=\{v_i\}_{i=1}^n$ with $n$ vectors where $v_i\in \mathbb{R}^d$ is a vector and a transfer matrix $\mathbf{W}\in \mathbb{R}^{d_1\times d}$, the target is to select two subsets $V_1=...
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Complexity of the distance between the average vector of two subsets

Given a vector set $V=\{v_i\}_{i=1}^n$ with $n$ vectors, where $v_i\in \mathbb{R}^d$ is a vector, the target is to select two subsets $V_1=\{v_j\}_{j=1}^{|V_1|} \subset V$ and $V_2=\{v_k\}_{k=1}^{|V_2|...
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Does the Christofides algorithm ensure this inequality?

Let $(X,d)$ be a finite metric space. Let $C$ be a Hamiltonian cycle (over $X$) outputted by Christofide's algorithm. Also, let $K$ be a minimum spanning tree. I am aware that Christofide's algorithm ...
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Multi-dimensional 0-1 Knapsack problem with a high number of dimensions

I would like to solve a multi-dimensional 0-1 Knapsack problem, by looking for approximation algorithms with constant approximation ratio if possible. Here the particularity is that the number of ...
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Maximize a special monotone submodular function - is it easier?

I am looking for a way to optimize the function $f$, defined below. First, fix some positive integer $k$ and let $c_1$ and $c_2$ be non-negative vectors in $\mathbb{R}^n$. Let $g$ be an increasing ...
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Online assignment lower bound results

I am reading the following paper which presents a $(1-\epsilon)$-competitive online algorithm for the MaxMin (similar to the makespan) problem, defined as follows: a set of requests are arriving in an ...
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2 votes
1 answer
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Another variation of $k$-means problem in the plane

According to wikipedia, consider $k$-means problem in the plane : k-means clustering aims to partition the $n$ observations into $k (≤ n)$ sets $S = \{S_1, S_2, \dots, S_k\}$ so as to minimize the ...
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Existing results around approximation of minimum 2-edge connected Steiner subgraph

Problem $1$: minimum 2-edge connected subgraph We are given a $2$-edge connected undirected graph $G(V,E)$, and we are asked to find a $\textit{spanning}$ subgraph $H$, with minimum number of edges, ...
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4 votes
2 answers
219 views

Maximum Vertex Cover

I recently encountered the following exam problem: Given an undirected graph $G := (V,E)$ and a natural number $k \geq 1$, we want to cover as many edges as possible using exactly $k$ vertices. ...
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Is logarithmic additive approximation to bin covering possible?

For the bin packing problem, the Karmarkar-Karp algorithm (1982) finds in polynomial time a packing with $OPT+O(\log^2(OPT))$ bins, and this was recently improved by Hoberg and Rothvoss (2017) to $OPT+...
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2-Center problem with forbidden pairs

Is there a nearly linear-time 2-approximation (or $O(1)$-approximation) algorithm for the following problem? 2-Center with Forbidden Pairs input: Bipartite graph $G=(V,E)$ where each vertex $v$ is a ...
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Hashing-based vs almost uniform sampling-based approximate counting

Corollary 3.6 in the UniqueSAT paper by Valiant and Vazirani [1] states: For any $\varepsilon > 0$ there is a randomized polynomial-time TM with a SAT oracle, which given a SAT formula $f$ outputs ...
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A counter example for the set mean objective

Let $\mathcal{P} = \{P_1, \cdots,P_n\}$ be a family of finite point sets in $\mathbb{R}^d$, each having at most $m$ points. Consider the following objective function \begin{align} cost(\mathcal{P},c) =...
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Minimize Cumulative Cost on Topological Sort

We are given a n-vertex DAG $G=(V,E)$ and also given a cost function $c: V \rightarrow \Bbb N$. Given a topological sort $S = v_1,v_2,...,v_n$, it has associated a sorting cost $S_c = \sum_{i=1}^{n} C(...
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Approximate solution for maximum coverage problem with choice constraint

Suppose a sequence of sets $S_1,S_2,...,S_i$ where each set contains sets of elements. That is, each set $S$ contains many sets $a_1,a_2,...,a_{|S|}$. We are given an integer $k$ and we assume that $\...
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3 votes
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102 views

Bin packing where each item must occur in $k$ bins

I am looking for information on a generalization of bin-packing in which each item should appear in exactly $k$ different bins, for some positive integer $k$. The standard bin packing problem ...
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Is there a primal-dual algorithm for the Tree Augmentation Problem or the Cactus Augmentation Problem?

The TAP problem and the CacAP problem can be seen as covering problems for the minimum cuts of a graph. It seems like these problems would fall under the framework of network design problems (...
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k-Median Problem With Restricted Centers

The $k$-median problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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1 vote
1 answer
197 views

State of the art approximation algorithm for $\text{MAXCUT}$ that does better than Goemans and Williamson

I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia: The polynomial-time approximation algorithm for Max-Cut with the best known ...
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Can this special case of Node Weighted Steiner Tree be solved in polynomial time?

Consider the node-weighted steiner problem: Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$. Output: a minimum weight subset $S \...
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2 answers
310 views

Partition a graph into two clusters

Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized? ...
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2 answers
214 views

Finding top-K items in a sliding window

Imagine we have a stream of bank transactions. Each transaction has a target account and some amount of money. I'd like to find top K accounts over some period of time (e.g. last 7 days) which ...
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Using bin-packing algorithms to approximate maximum-makespan

Bin-packing (BP) and maximum-makespan (MM) are dual problems. In both problems, the input can be defined as a set $S$ of positive integers, and the output is a partition of $S$. In BP, there is a ...
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1 answer
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Does an upper bound on the integrality gap imply an approximation algorithm with the same ratio?

Often, we can model combinatorial optimization problems with an Integer Program. Then there is an associated Linear Relaxation which drops the integrality constraints on the variables. Let's say we ...
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Examples of SDP constant approximation algorithms on minimisation problems

I was recently going through a survey on semidefinite programming and its use in approximation algorithms. Here are some problems I am familiar with that have SDP approximations: Max Cut ($\approx 0....
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Failing to understand a lemma regarding Robust Low Rank Approximation

I am reading Low Rank Approximation in the Presence of Outliers by Bhaskara and Kumar and kind of stuck at the proof of Lemma 9. The paper studies robust (to outliers) low rank approximation problem. ...
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Modifying sets to minimize the distance among each pair of the mean value of sets

Given $n$ points, each point $x_i$ has a value $v_i \in \mathbb{R}^{d}$, and there are $m$ point sets $\{S_1,\dots, S_m\}$ that each point set consists of some points. The size of point sets can be ...
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1 answer
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Set cover where consecutive sets differ by at most one item [closed]

First I define my version of the set cover problem: We have a collection of sets such as $S_1, \dots, S_m$ where each $S_i$ is a subset of $M=\{1,\dots, m\}$. The goal is to find the minimum number of ...
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1 vote
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A variant of k-median clustering

Suppose $\mathcal{P} =\{P_1,\cdots,P_n\}$ is a family of $n$ finite sets in $\mathbb{R}^d$. Given set $C=\{c_1,\cdots,c_k\}$ of $k$ points, consider the follwoing objective funtion $cost(\mathcal{P},C)...
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5 votes
1 answer
238 views

$\rho OPT + k$ approximation for bin packing (Unpublished result of David P. Williamson)

I am currently stuck on Exercise 5.12 in this book, which is apparently an unpublished result of David P. Williamson according to the book notes. The problem asks to use randomized rounding and first ...
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2 votes
2 answers
330 views

Is there any Bi-criteria PTAS for Metric $k$-Median?

The $k$-median problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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-2 votes
1 answer
315 views

Find research partner (profession and beginner)

I've 10 years of industrial work, but in my free time, I do research, write papers to conferences, help to teach to my old friend at the university and I even did a Ph.D. full-time program. Now, I've ...
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2 votes
1 answer
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Do there exist two equivalent objective functions one of which can be approximated but another one cannot?

I have two equivalent problems A and B, meaning that the optimal solution of one must be the optimal solution of another one. However, it seems that problem A can be approximated but B cannot. Below ...
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3 votes
0 answers
124 views

Incorrect Lower Bound of k-Means++ Algorithm

The $k$-means++ algorithm is composed of two parts: Initialization part: the initial $k$ centers are chosen based on $D^2$ sampling. Expectation maximization part: the standard $k$-means algorithm (...
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3 votes
1 answer
111 views

Approximating Independent Dominating set on bipartite graphs

I'm interested in the following problem: given a bipartite graph, find the smallest independent set of vertices which dominate all other vertices. My question is: are there any positive results in the ...
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1 vote
2 answers
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Two valued variant of subset sum problem

I'm interested in the complexity of the following problem: Given a multiset $S$ containing only two positive integers $a$ and $b$, find a $k$-partition of $S$ that maximizes the sum of part with ...
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2 votes
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Is there a name for approximation algorithms with $f(OPT)$ approximation factor?

I have read that there are approximation algorithms with two different kinds of approximation factors: $c$ . A constant approximation factor. $f(n)$ . An approximation factor that is function of the ...
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3 votes
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154 views

Improving the approximation in Stockmeyer's counting theorem

Given a $\#P$ function $f(x)$, we can use Stockmeyer's counting theorem to get an approximation $g(x)$ such that \begin{equation} \left(1 - \frac{1}{\text{poly}(n)} \right) f(x) \le g(x) \le \left(...
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1 answer
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Minimum Degree Spanning Tree Without Restricting Vertices Searched

This is reposted from cs.stackexchange. I asked the question more than two weeks ago and got no answers, so I thought to repost here. I am currently self studying approximation algorithms from The ...
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1 vote
1 answer
62 views

What is the run-time of the bin packing approximation algorithm?

The best approximation algorithm that I found for the bin packing problem is by Hoberg and Rothvoss (SODA, 2017). In their Theorem 1.2, they mention that their algorithm finds a solution with at most $...
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10 votes
1 answer
415 views

Is subtractive dithering the optimal algorithm for sending a real number using one bit?

Consider the problem of sending a real number $x\in[0,1]$ using a single bit $X\in\{0,1\}$ in an unbiased manner. We assume that the sender and receiver have access to shared randomness $h\sim U[-1/2,...
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3 votes
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Hessian of non differentiable convex function

The motivation of the question is the following: Let $P$ be a set of $n$ points in $\mathbb{R}^d$. Consider the following objective(convex and differentiable) function $f:\mathbb{R}^d\rightarrow [0,\...
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1 vote
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113 views

Hardness of Approximation of Continuous Metric k-Median

First let me describe the metric $k$-median problem. Definition (Metric $k$-Median): Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ ...
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1 answer
218 views

A k-approximation to k-way number partitioning

The $k$-way number partitioning problem accepts as input a multiset $S$ of positive numbers, and returns a partition of $S$ into $k$ subsets such that the subset sums are as nearly-equal as possible, ...
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