# Questions tagged [approximation-algorithms]

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### Practically Good Algorithms of a Very Low Computational Complexity Class

I am looking for one (or more) examples of a parametric class of algorithms $P_t$ for approximately solving a class $\cal A$ of algorithmic questions with the following properties: 1) Solving the ...
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### Complexity of approximating the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}|.$$ I believe that it is NP-hard to compute $S_M$ exactly, by applying ...
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### Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)

Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same. First, I will define a few ...
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### 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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### Statistical Algorithms vs Convex Relaxations - Planted Clique

I am trying to understand exactly what the lower bounds for the query complexity of statistical algorithms imply for convex relaxations for the planted clique problem ? A recent paper by Feldman, ...
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### Bipartite vertex separator

Are there any common approaches for finding a vertex separator in a bipartite graph $G = (V_1, V_2, E)$ where the selected vertices are constrained to come from one partition of the graph? I have a ...
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### On the optimal solution of the CKR formulation for MULTIWAY CUT

Currently the best approximation algorithm for the MULTIWAY CUT problem is obtained via the linear program based on geometrical embedding by CKR . Let $U_i$ be those vertices in $V-T$ which is ...
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### A question regarding Improved Algorithm for Degree Bounded Survivable Network Design Problem

In the paper "Improved Algorithm for Degree Bounded Survivable Network Design Problem", by N. Vishnoi and A Louis, have used the iterated rounding approach in a similar as by Jain in designing the ...
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### Finding an index set so that row sums are positive

Assume $A$ is a $n$-dimensional matrix of real numbers. The diagonal entries are non-negative, and all other entries are non-positive. I would like to find a subset $I \subseteq \{1, 2, \ldots n\}$ of ...
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### Hardness of approximating chromatic number of triangle-free graphs

The chromatic number of graph, $\chi( G)$ is hard to approximate for general graphs. Are there results of hardness of approximating $\chi(G)$ for triangle-free graphs?
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### $XP_{\text{uniform}}=FPT$ and update to $EPTAS$ section in complexity zoo?

Complexity zoo in https://complexityzoo.uwaterloo.ca/Complexity_Zoo:E#eptas has the following: $FPT = XPuniform\implies EPTAS = PTAS$. Fundamentals of Parametrized complexity on page $534$ has ...
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### Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
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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 ...