Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

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30 views

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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48 views

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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1answer
160 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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60 views

A Special case of Weighted Set Cover Problem

We are given a set P of k points in the plane, and a set S of simple polygons that are triangulated into a set T of triangles, each triangle t in s has a weight that is 1 if we choose no triangles (w(...
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1answer
72 views

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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2answers
240 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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1answer
119 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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42 views

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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1answer
114 views

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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58 views

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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49 views

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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40 views

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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44 views

Low Rank Approximation of a hidden subset

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $Q\subseteq P$ with $\vert Q\vert \geq \alpha n$ for some constant $\alpha\in(0,1]$. Given a $j$-dimensional affine subspace(flat) $F$ consider the ...
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55 views

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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38 views

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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189 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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2answers
249 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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1answer
230 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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1answer
120 views

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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115 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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97 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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2answers
189 views

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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65 views

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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101 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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1answer
122 views

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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25 views

Given an input sequence of real numbers, how to find the closest sequence in a large set of sequences

We are given a set $S$ of $m\gg 1$ sequences (arrays) of $n$ elements, where each sequence $s\in S$ belongs to $\mathbb{R}^n$. In the problem I am trying to solve, in a sequential fashion, we obtain a ...
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1answer
47 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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372 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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41 views

are there approximation algorithms that use primal-dual with LP values and/or rounding?

Are there approximation algorithms that use primal-dual with LP values and/or rounding? e.g. An algorithm that during any iteration first tries to see an extreme point to the LP has any value above a ...
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68 views

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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88 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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48 views

Balanced and general $MAXkSAT$ known approximation results and bounds from $UGC$

$MAX2SAT$ has a $0.9401$ to $0.9402$ approximation algorithm which is conjectured to be optimal by $UGC$ while there is a balanced $MAX2SAT$ bound of $0.943$ approximation which is conjectured to be ...
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1answer
153 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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1answer
135 views

Is this homework problem on T-joins wrong? [closed]

In Question 9.3a, it states that if $T=V$, then the minimum cost perfect matching is the minimum cost T-join. Is this actually true? I think I have a counterexample which I have drawn below.
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A continuum version of the 1D k-means clustering problem: constant factor approximations

Modify the k-means clustering problem in 1D by assuming that, instead of a finite number of observations, we must classify into $K$ clusters a continuum of observations, distributed on the unit ...
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1answer
52 views

Best approximations of Minimum Dominating Sets in chordal graphs

I am searching results and papers related to the (in)approximability of the Minimum Dominating Set problem in chordal graphs. In particular, what is the best approximation ratio achievable in polytime ...
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114 views

Algorithms and approximations for optimal offline binary tree operations

Let's say we are using a binary tree to represent a set of elements, with operations $\mathsf{insert}(x)$ and $\mathsf{delete}(x)$. We will assume that the operations are used such that a deleted ...
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97 views

Can hash functions speed up quantum simulation? (Generalizing May and Schlieper's idea) [closed]

To conform with the CS Theory SE crossposting rules, I've created a separate post for dequantizing Shor's algorithm (discussion on the Quantum Computing Stack Exchange was mostly about Shor's ...
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90 views

Jump number approximation algorithm

A linear extension $x_1 x_2 \ldots x_n$ of a partially ordered set (poset) is said to have $k$ jumps if there are $k$ occurrences of consecutive elements that are incomparable with each other -- i.e., ...
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67 views

Can the theory of Bidimensionality be applied to weighted instances of a problem?

So my understanding of bidimensionality is you are assured the problem solution is about O(k^2) so you can pay O(k) purely to reduce the instance to one of bounded treewidth. As far as I know, this ...
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54 views

polytime approximability of directed multicut

Does anyone know how well directed multicut can be approximated in planar and minor free graphs? Also any survey of approximability of directed multicut and multicut in various graph classes would be ...
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94 views

approximate maximum clique given vertex cover

I have a non optimal vertex cover of size k of a graph G, and I want to get a (1+epsilon)-approximation kernel of size linear in k for maximum clique of G. One thing I got is that every clique in G ...
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2answers
107 views

Where to find info on (polytime) approximability of various discrete optimization problems?

Where to find info on (polytime) approximability of various discrete optimization problems? Sorry if this is stupid,but is there a site or reference that keeps up to date info on approximability of ...
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142 views

Complexity of approximating a real function using queries

Consider the following computational problem, where $I$ is the real interval $[-1,1]$: There is a monotonically-increasing function $f: I\to I$. You are allowed to access it only through queries of ...
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1answer
162 views

What are the worst-case and average-case time complexities of the greedy algorithm for the weighted set cover problem?

Let $X$ be the universe of elements, $F$ a collection of subsets $S \subseteq X$, each with an associated cost. The goal is to find a subcollection $C \subseteq F$ of minimum total cost which covers $...
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1answer
214 views

Good Survey paper for k-means/k-median/k-center/facility-location

I have stated 4 problems in the Question title. All these problems are closely related and are studied in various variations. For example: Space: Euclidean/metric/discrete/continuous/non-metric/2-...
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1answer
526 views

Is the following graph optimization problem approximable within a constant factor?

Let $G=(V,E)$ be an undirected graph, and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{pred}_{\pi}(v)$ (respectively $\text{succ}_{\pi}(v)$) the set of ...
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95 views

Separation oracle for hitting all small cut on a graph?

We are given as input an undirected graph $G=(V,E)$, weights $w_e \ge 0$ for all $e\in E$ and an positive integer $k$. We aim to select a set of edges with the minimum weight, such that the cut set of ...
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2answers
174 views

kmeans++ for arbitrary metric spaces and general potential function

I was reading this popular paper "k-means++: The Advantages of Careful Seeding". It appeared in SODA 2007. Since this technique is the most popular clustering technique, I am hoping that my question ...
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34 views

Different version of approximation complexity and algorithm for densest-k-subgraph problem

In the densest-k-subgraph problem, we are given a graph $G =(V,E)$ and $k$, and we are asked to find a set $S \in V$ of vertices to maximize the number of edges in the induced graph of $S$, i.e. $|Ind[...

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