# Questions tagged [approximation-algorithms]

An approximation algorithm is an algorithm that finds an approximate solution to a (typically NP-hard) problem. The quality of the algorithm is measured by how close to the actual optimum it performs. For example, it is a constant factor approximation algorithm if it always outputs a solution that is within a constant factor of the optimum. Hardness of approximation is one way to separate NP-hard problems.

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### Longest path on directed acyclic graph when the weight is defined on the pair of edges

Given a directed acyclic graph $G=(V,E)$ with a source node $s$ and a sink node $t$, and we have a weight function that is defined on $E\times E$, $f:E\times E\to R^{+}$. We want to find a $s$-$t$ ...
122 views

### How to find an optimal sequence of merging operations?

Given a set of items, each characterized by a quality $q_i\in(0,1)$. We can merge two items of quality $q_i$ and $q_j$ to a single item $k$ of quality $q_k=f(q_i,q_j)$, where $f$ is increasing in $q_i$...
1 vote
116 views

### Steiner tree subject to non-trivial constraint

Given a edge-weighted transportation network modeled as a graph. A source node $s$ needs to send an object to a set of $k$ destination nodes $t_i$, $1\le i\le k$. For the transportation, $s$ needs to ...
44 views

### Maximize connectivity probability with a number of edges

We are given a graph $G$, whose edges are either open or closed. Initially all the edges are closed. For each edge $e$, if we choose to activate it, then after the activation, it becomes open with ...
52 views

### Graph reduction and combinatorial optimization

Crossposted at Theoretical Computer Science SE We are given a multigraph $G$. Consider two nodes $u$ and $v$ with multiple edges between them. Each elementary edge is associated with a metric called ...
75 views

### Is identifying the best in randomly chosen $n$ elements, equivalent to identifying one from the best half of randomly chosen $2n$ elements?

Suppose we are given a set $U$, and a black-box objective function $f: U \mapsto [0, 1]$. The job is to maximise $f(\cdot)$. Now, for a given $\delta \in (0,1)$, consider the following randomised ...
1 vote
52 views

### Steiner tree subject to edge capacity constraint

Given a network of routes modeled as a graph where each edge $e$ has a capacity $c_e$. We have a source node $s$ and a set of destination nodes $t_i$ ($1\le i\le k$). We need to transport $q_i$ ...
1 vote
40 views

### Evaluate the goodness of piecewise linear approximation of a cubic Bézier [closed]

I saw that there are different algorithms for piecewise linear approximation of cubic Bézier curves out there. Now suppose that these algorithms can be either used interchangeably or that the ...
150 views

### Metric TSP with integer edge cost

Given a metric TSP with integer edge cost upper-bounded by a constant $C_{\max}$, can we find an poly-time algorithm solving this TSP instance?
87 views

### What is the complexity of a special multigraph edge coloring problem

Given a multigraph such that there are 0 or 2 edges connecting every two vertices, we are to color the edges of this graph so that adjacent edges receive distinct colors. It is known that we need at ...
34 views

### Approximabilty of submodular over modular maximization

Given a non-decreasing, normalized, submodular function $f : 2^{[n]}\mapsto \mathbb{R}_+$ and a modular non-decreasing function $g$, I am wondering what is the best approximation ratio I can hope for ...
172 views

### Approximation of integral of gaussian function over a parallelepiped

Remark: I posted this question in math stackexchange here and computer science stackexchange https://cs.stackexchange.com/ few weeks ago but obtain no answer. Given a multi-dimensional gaussian ...
1 vote
37 views

### Computational complexity of rate $\frac{1}{2}$ codes

We know from Berlekamp, McEliece and Van Tilborg [On the inherent intractability of certain coding problems, IEEE Trans. Information Theory, 24 (1978)] that computing the minimum distance of a (binary)...
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1 vote
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### How to compress variables in a linear regression

I have a large linear regression where all the independent variables are logical (ie TRUE/FALSE) and sparse. The data has roughly 10,000 variables and 10 million observations, on average around 20 ...
123 views

### Can one optimize the probability that an identity is satisfied until the probability is $1$?

I wonder how well one can obtain algebraic structures that satisfy interesting algebraic identities by simply slowly modifying those algebraic structures until they satisfy the required identities. I ...
116 views

### complexity of bounded knapsack with spoilage

Consider the usual bounded knapsack problem, with the extra twist: you know that $k$ of the chosen items will get spoiled after the sack is packed. And this happens adversarially, i.e. $k$ most ...
285 views

### Approximate homology of a large simplicial complex

I can use software to calculate the Betti numbers $\beta_0,\beta_1,\beta_2,\dots$ of a finite simplicial complex. This is prohibitive for large complexes, built on say > 100,000 nodes. Is there some ...
643 views

### Approximation of half-integers modified Bessel function of the second kind

I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...
346 views

### Counting spanning trees of a planar graph

I know through Kirchoff's Theorem, one can calculate the number of spanning trees via the determinant of a Laplacian. This has complexity $O(N^{2.373}$). I was wondering if anyone was aware of a ...
303 views

### The minimum rank of a matrix over GF(2) when part of non-zero off-diagonal elements are set to be zero

Given an $n\times n$ matrix $A$, whose elements are over $GF\left(2\right)$ and all diagonal elements are $1$. There are $m\ (m\leq n^2-n)$ non-zero off-diagonal elements in $A$. If we are allowed to ...
42 views

### A name for algorithms that perform well in an asymptotic sense, when inputs are random

Is there a term for an algorithm that performs "well" (say, within a constant factor of optimality) in an asymptotic sense when a large number of random inputs are provided? For example, say I had an ...
225 views

### Fast Algorithms for sum of independent random variables

CLT implies the sum of n i.i.d random variables,after property normalized converge to a Normal distribution as n goes to infinity. Furthermore, Linderberg's condition points out not necessarily ...
1 vote
904 views

### Removing the minimum number of edges to make a graph triangle-free (using set cover)

Assume that we are given a weighted, undirected graph $G = (V; E)$ where each edge $e \in E$ is assigned weight $w(e) \geq 0$. The goal is to remove a set of edges $D \subseteq E$ with minimum weight ... 264 views

### What is the LP gap of vertex cover in planar graphs?

What is the LP gap of vertex cover in planar graphs? The LP I refer to is min $\sum_{e \in E } c_e x_e \ \$ subject to $\ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E$ $c_e \geq 0$ are ...
1 vote
Let $D=(V,A)$ be a simple directed acyclic graph, where $A$ is a set of arcs. Let $S$ be a subset of $\{(u,v)| \text{there is a directed path from$u$to$v$}\}$. The $S$-length of a path $P$ is ...
Are there any known heuristics for the following variation of the traveling salesman problem: given $n$ sets of points $S_1,\dots,S_n$, and $n$ integers $k_i$ such that $k_i \leq |S_i|$, find the ...