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Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

6
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1answer
2k views

What's the approximation factor of this Max k-Cut approximation?

I'm thinking about an approximation algorithm for Max k-Cut. One simple and another one advance approximation algorithms are available here. The Max k-Cut problem is defined as follow: Assume we ...
16
votes
3answers
453 views

Why differential approximation ratios are not well-studied comparing to standard ones despite of their claimed benefits?

There is a standard approximation theory where the approximation ratio is $\sup\frac{A}{OPT}$ (for problems with $MIN$ objectives), $A$ - the value returned by some algorithm $A$ and $OPT$ - an ...
7
votes
1answer
675 views

Approximating transitive reduction of a transitive closure of a dag

Let's suppose a transitive closure $G^+$ of a dag $G$ is given and we want to compute an approximation of the transitive reduction $G^-$ such that the full transitive reduction is a subgraph of the ...
11
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1answer
668 views

Why do NP-complete problems not have similar approximation ratios?

Since 2 NP-complete problems are by definition reducible to each other, so a solution to one of them can be obtained by using a black-box solving the other one, why don't they have similar ...
5
votes
3answers
428 views

Partitioning a segmented stick

Problem : We are given a stick partitioned into n - equal parts. Each of these parts has a weight, let's say x. Number of times x appears as weight of some part is guaranteed to be even. For ...
3
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2answers
483 views

Is there parallel algorithm for 3SAT

Is there any parallel algorithms or approximation algorithms for 3SAT?
48
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3answers
2k views

Is there a sensible notion of an approximation algorithm for an undecidable problem?

Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be ...
22
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3answers
574 views

Educational Source or Survey on Analysis of Semidefinite Program?

When designing approximation algorithms one sometimes solves a semidefinite program followed by a rounding step. An often used example to illustrate this is Max-Cut. (See e.g. Approximation Algorithms ...
7
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3answers
2k views

A simple approximation algorithm for the TSP

Consider the following extremely simple approximation algorithm for the TSP. Input: A complete weighted graph $G=(V,E).$ Take any three vertices $a,b,c\in V$ and let $H:=(a,b,c,a).$ While there ...
12
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3answers
610 views

A Multi-cut Problem

I'm looking for a name or any references to this problem. Given a weighted graph $G = (V, E, w)$ find a partition of the vertices into up to $n = |V|$ sets $S_1,\ldots,S_n$ so as to maximize the ...
17
votes
3answers
429 views

Is there a constant factor approximation algorithm for 2D rectangle coloring problem?

The problem we consider here is the extension of the well-known interval coloring problem. Instead of intervals we consider rectangles having sides parallel to axes. The objective is to color the ...
12
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3answers
534 views

What are the problems with the best approximation ratio achieved by algorithm returning uniformly random solution?

What are the problems with the best known approximation ratio achieved by an algorithm returning a uniformly random solution? I know one such example for permutation flow shop problem $F|perm|C_{max}...
7
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2answers
658 views

Approximation algorithms for MAX-CUT, when sizes of partition sets are fixed

The MAX-CUT problem has constant factor approximation, but we can't control the sizes of the sets in resulting partition. What is known about maximizing cut size, if we restrict one part of the ...
16
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2answers
695 views

Set Cover for Permutation Matrices

Given a set S of nxn permutation matrices (which is only a small fraction of the n! possible permutation matrices), how can we find minimal-size subsets T of S such that adding the matrices of T has ...
10
votes
1answer
204 views

Relaxing $\ell_0$ constraints in an optimization

I have a feasibility question that can be framed as follows. I'm given a point $p$ in a $d$-dimensional vector space, and I want to find the closest point $q$ to $p$ that satisfies a set of "$\ell_0$ ...
15
votes
1answer
418 views

Graph decompositions for combining “local” functions of vertex labelings

Suppose we want to find $$\sum_x \prod_{ij \in E} f(x_i,x_j)$$ or $$\max_x \prod_{ij \in E} f(x_i,x_j)$$ Where max or sum is taken over all labelings of $V$, product is taken over all edges $E$ for a ...
12
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3answers
3k views

Is there an online-algorithm to keep track of components in a changing undirected graph?

Problem I have an undirected graph (with multi-edges), which will change over time, nodes and edges may be inserted and deleted. On each modification of the graph, I have to update the connected ...
19
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1answer
317 views

What are the best possible time/error tradeoffs for approximate solution of linear programs?

For concreteness consider the LP for solving a two-player zero-sum game where each player has $n$ actions. Suppose each entry of the payoff matrix $A$ is at most 1 in absolute value. For simplicity ...
34
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3answers
3k 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 ...
8
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2answers
688 views

Approximation Algorithms for MAXSAT

Trying to find the optimal solution to WEIGHTED-MAX-3SAT, the weighted version of the 3-SAT optimization problem, is NP-hard. In fact, even approximating the non-weighted version of MAX-SAT ...
4
votes
1answer
407 views

Better approximation for special case of 3-hitting set

I have a question based on 3-Hitting Set problem. In this problem, we are given a universal set U of size n and a set of subsets S such that $\forall $ s $\in$ S |s|<=3. FOr this problem, Integer ...
4
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0answers
221 views

Follow-up on Nair/Tetali's Correlation Decay Tree for hypergraphs?

I'm wondering if anybody has followed up on Nair/Tetali's correlation decay tree construction for hypergraphs. I didn't find anything relevant in back-citation on google scholar. Interesting question ...
25
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3answers
912 views

When is relaxed counting hard?

Suppose we relax the problem of counting proper colorings by counting weighted colorings as follows: every proper coloring gets weight 1 and every improper coloring gets weight $c^v$ where $c$ is some ...
7
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2answers
892 views

Integer Factoring via Lattice Reduction?

I found a paper titled "Factoring integers and computing discrete logarithms via diophantine approximation" by C. P. Schnorr from 1993. It looks like a probabilistic method with expected polynomial ...
21
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2answers
1k views

Polynomial time approximation algorithms for machine scheduling: how many open problems are left?

In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper "Polynomial time approximation algorithms for machine scheduling: Ten open problems". Since then, to the best of my knowledge, ...
38
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9answers
3k views

Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
11
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2answers
455 views

Average distortion embeddings

Consider two metric spaces $(X, d)$ and $(Y, f)$, and an embedding $\mu : X \rightarrow Y$. Traditional metric space embeddings measure the quality of $\mu$ as the worst-case ratio of original to ...
14
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1answer
875 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
11
votes
4answers
454 views

Dimensionality reduction with slack?

The Johnson-Lindenstrauss lemma says roughly that for any collection $S$ of $n$ points in $\mathbb{R}^d$, there exists a map $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ where $k = O(\log n/\epsilon^2)$ ...
5
votes
1answer
399 views

High probable polynomial time algorithm for NP-hard problems?

Many NP-complete or NP-hard problems are addressed under the assumption that the problem instances are uniformly distributed. However, it is not true for real world applications. One example is the ...
8
votes
2answers
340 views

complexity of fitting models to data

Suppose $f:\mathbf{R}\times \mathbf{R} \to \mathbf{R}$ is some some continuous function $x_1 \ldots x_n$ is a set of real values, and we'd like to compute $\text{argmin}_a \sum_i f(a,x_i)$ to ...
11
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2answers
3k 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 ...
44
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8answers
5k 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 ...
24
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4answers
2k views

Bounded-cardinality bounded-frequency set cover: hardness of approximation

Consider the minimum set cover problem with the following restrictions: each set contains at most $k$ elements and each element of the universe occurs in at most $f$ sets. Example: the case $k = 4$ ...