Questions about approximation algorithms.

learn more… | top users | synonyms

3
votes
0answers
28 views

Approximation of covering number in metric space

Consider the following setting: Let $(X,d)$ be a metric space and let $S$ be a finite subset of $X$. An $\epsilon$-cover of $S$ is any subset $C\subset S$ such that $$ \max_{x\in S} d(x,C)\leq ...
6
votes
1answer
175 views

Why is complementary slackness important?

Complementary slackness (CS) is commonly taught when talking about duality. It establishes a nice relation between the primal and the dual constraint/variables from a mathematical viewpoint. The two ...
3
votes
1answer
1k views

A quantum algorithm for GCD

Does anyone know of a direct quantum algorithm for computing GCD, - There could be quantum gates for addition subtraction constructed explicitly, using CNOT, etc. - the construction can be done in ...
0
votes
0answers
45 views

How can I show that zero-one programming is not in APX?

How can I show that zero-one programming is not in APX? Vertex Cover Problem is in the APX class. So can I try a PTAS-reduction from the zero-one programming problem to Vertex Cover and show that ...
8
votes
0answers
187 views

Additive error in counting the number of 1's in a sliding window?

The setting is as follows: We're given a stream of bits. At time $t$ you get to see bit $b_t$, and required to output $\widehat{s_t} \approx \Sigma_{i=0}^{N}b_{t-i}$ (i.e. approximately how many 1's ...
1
vote
2answers
69 views

Maximum item in data stream

I have N elements and need to find the maximum of these elements. At each time tick, exactly one of the N elements is updated and I need to determine the new max element (more specifically, the index ...
0
votes
0answers
52 views

Estimate the maximum by randomized approximation algorithms

Given a set $A$ of $2^n$ elements such that each $a\in A$ is a rational in $[0,1]$. The question is to estimate $\max(A)$, i.e., the maximal element of $A$. Is there a polynomial algorithm to ...
9
votes
1answer
148 views

Bisecting a set of points into two optimal subsets

I want to divide a set of points into two equally-sized subsets such that the within-cluster sum of squares is minimized. We can assume that the points are in two-dimensional Euclidian space. I'm ...
8
votes
2answers
157 views

Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
3
votes
3answers
143 views

The number of integral points in a polytope

Suppose we define a polytope with $$ \mathbf{Ax} \leq \mathbf{b} $$ What is the best way to find/approximate the number of the integral points in the polytope? Update: how hard is the complexity ...
0
votes
0answers
120 views

Calculating exact/approximate solution to a formula

Suppose we have a set of variable $\mathbf{y} = \left(y_1, ..., y_n \right)$. Also consider the set of functions $g_i(y_i), 1 \leq i \leq n$. Note that $g_i()$ is dependent only on $y_i$. Consider ...
2
votes
1answer
91 views

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?
5
votes
0answers
113 views

Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
7
votes
1answer
166 views

Tight examples for approximating the feedback vertex set problem

There are several 2-approximation algorithms for the UNWEIGHTED feedback vertex set problem (FVS), which are summarized in [4]. Note that the reduction from vertex cover to FVS is ...
1
vote
0answers
29 views

Approximation algorithm to a minimization problem with optimization function going to zero

I have a simple function to minimize, but it is not discrete: $f(x) = 2x - \alpha x - c(x)$, where $x$ is a natural number, $\alpha$ is a rational number, $0 \leq \alpha \leq 1$, and $0 \leq c(x) \leq ...
6
votes
1answer
78 views

Minimize makespan on identical machines when jobs are vectors

Given are $n$ $d$-dimensional vectors and $m$ machines where $d$ need not be fixed. The objective is to minimize the makespan i.e., assign the vectors to machines such that the maximum of the ...
14
votes
1answer
317 views

Does PSPACE-completeness imply approximation hardness?

It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it? Is this "tight"? (i.e., are there ...
7
votes
2answers
169 views

Planted Clique in G(n,p), varying p

In the planted clique problem, one must recover a $k$-clique planted in an Erdos-Renyi random graph $G(n,p)$. This has mostly been looked at for $p=\frac{1}{2}$, in which case it is known to be ...
2
votes
1answer
62 views

State of the art on approximating quadratic assignment problem

I was wondering what is the state of the art on approximating the quadratic assignment problem (QAP). In particular, I am interested in the following instance. Suppose the $A = (a_{ij}) \in \{0,1\}^{n ...
4
votes
1answer
95 views

Polynomial-time distinguishability threshold of planted clique

I have a basic question regarding the best known polynomial-time "distinguishing advantage" for the planted clique problem. By this, I'm referring to the problem of distinguishing the distribution ...
4
votes
1answer
101 views

Gradient descent-like optimization on a convex landscape with noisy sampling

We have a strictly convex function $f(x,y)$ with a global minimum at $p_{min}$. The goal is to approximate the minimum. E.g. $$f: [0,\pi]^2 \to \mathbb{R}$$ $$f(\theta,\phi) = t_1 \sin \theta + t_2 ...
3
votes
2answers
206 views

Greedy MAX SAT approximation ratio

Consider a naive MAX SAT approximation algorithm: pick a literal $l$ which appears in maximum number of clauses set the corresponding variable of $l$, such that all clauses containing $l$ are ...
2
votes
2answers
148 views

Set packing with maximum coverage objective

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. Set-Packing asks how many disjoint sets we can pack, and is defined ...
1
vote
1answer
80 views

Covering by disjoint sets

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. I'm interested in the approximability of two problems, or in ...
4
votes
1answer
56 views

Bellman principle and approximability

Does anybody know if a combinatorial optimzation problem that enjoys the Bellman's optimality principle can in automatic way be approximated?
6
votes
2answers
303 views

Proof for ACYCLIC PARTITION being NP-complete

I'm new to this site, so please pardon me for any mistakes and please feel free to edit the question to help get better answers. I'm interested in reading any proof of ACYCLIC PARTITION (Garey and ...
7
votes
1answer
174 views

How bad can the greedy coloring (list color) for the c-chromatic number of graph be?

c-chromatic number is defined in the paper Partitions of graphs into cographs. It asks for the minimum number of colors used to color vertices such that each color class is a cograph. Cograph is a ...
13
votes
1answer
254 views

Is DAG subset sum approximable?

We are given a directed acyclic graph $G=(V,E)$ with a number associated with each vertex ($g:V\to \mathbb{N}$), and a target number $T\in \mathbb{N}$. The DAG subset sum problem (might exist under a ...
0
votes
3answers
164 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
2
votes
0answers
177 views

Why doesn't the standard analysis of set cover $H_n$ greedy extend to partial cover?

Several authors, starting with Slavik, have noted that the classical analysis of the set cover $H_n$ greedy algorithm does not readily extend to the set partial cover problem, where the goal is to ...
1
vote
1answer
110 views

FPRAS for Perfect Matching

If you have FPRASes for counting number of matchings of size $\leq n$ and size $\leq n-1$, can you get an FPRAS for counting number of matchings of size $n$ (i.e perfect matchings)?
1
vote
0answers
53 views

Bound the number of rounds in the sampling

Suppose we have a sequence $a_1,a_2,\ldots, a_n$, each $a_i$ is sampled uniformly and independently from $[0,1]$. Define $$ J_1=1,\\ \text{for}~i>1, ~J_i = 1 \iff a_i < \min ...
5
votes
1answer
93 views

Worst case ratio between minimum clique cover and maximum independent set

The maximum independent set problem gives a lower bound for the minimum clique cover problem. This is easy to see because given any clique cover together with an independent set, any two vertices ...
1
vote
0answers
54 views

Sync time between two points

Suppose we have two points, A and B, each point has its own clock. We are able to send messages between ...
3
votes
0answers
80 views

Practical Implications of Kolmogorov's Result on the Universal Approximation Theorem with Neural Networks

After having read matus's beautiful answer in this thread explaining (among other things) Kolmogorov's result regarding the Universal Approximation Theorem with Neural Networks, I wonder: if just ...
1
vote
0answers
48 views

Generalization Issues with Practical Suggestions from Universal Approximation Theorem with Neural Networks

After having read matus's beautiful answer in this thread explaining (among other things) Cybenko's proof of the Universal Approximation Theorem for Neural Networks, I wonder: if we use a piecewise ...
3
votes
0answers
65 views

Sub optimal regex equivalence

Regex Equivalence is a hard problem which in general takes exponential space and exponential time. Are there any approximation/sub-optimal algorithms with some theoretical guarantees over equivalence ...
5
votes
1answer
287 views

Edge and vertex fault tolerance in graphs

Suppose we are given two graphs $G$ and $H$, where $H$ is a subgraph of $G$. What is the maximum number $k$ such that if any $k$ edges are removed from $G$, $H$ still remains a subgraph of $G$? What ...
3
votes
2answers
176 views

Variants of Densest Subgraph Problems

Given a graph $G$, and a vertex-induced subgraph $H$ of $G$, there are two superficially-similar definitions ways to define the density of $H$: (1) Average degree of vertices in $H$ (2) What I will ...
3
votes
1answer
112 views

Adaptive vs Bernoulli sampling

Flajolet analyzed Wegman's Adaptive Sampling for estimating distinct values (species) in a stream (population), giving an unbiased estimator for mean and specified the standard error to be ...
1
vote
1answer
83 views

Multi-Terminal Cuts - Applications and Verifying validity of an approximation algorithm

First thing first - I am not a CS guy. I am EE student (Systems and Signals). So it would help if you didn't use any big words :) The Multi-terminal cut the input is a graph G and a subset T of its ...
8
votes
0answers
121 views

Approximating a convex polyhedron, with fewer inequalities

I have a convex polyhedron $\mathcal{P}$, given by $n$ linear inequalities $a_i \cdot x \le c_i$ where $x$ is a $d$-dimensional vector over the non-negative real numbers. In other words, ...
3
votes
1answer
93 views

Permanent Approximation - Why can the JSV algorithm not handle matrices with negative entries?

Going through the literature, it seems that what it comes down to is that if one could efficiently approximate permanents of matrices with negative entries, then that would imply an efficient ...
3
votes
2answers
647 views

Top-k frequent items in data stream

Paper "HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm" introduce an efficient algorithm which can help to estimate distinct items in large data set. I am thinking about ...
1
vote
0answers
76 views

Tiling a rectangle with weighted cells (min-max problem)

Given a sparse matrix $A[1..n, 1..n]$ containing cells with integer value of either $0$ or $1$, partition it in $J$ non-overlapping axis-parallel 2D rectangles, such that each $1$-cell is covered by ...
14
votes
2answers
248 views

Approximation in subexponental time

There are studies about approximation algorithms for NP complete problems in Polynomial time and exact algorithms in exponential time. Are there studies about approximation algorithms for NP complete ...
4
votes
0answers
154 views

Size of Independent set of sparse graphs with few triangles

Notations $\alpha(G) = $ Max sized independent set of graph $G$. $n(G) = $ Number of vertex in graph $G$. Theorem (by Ajtai et al.): For a triangle-free graph $G$ and max degree being $\Delta$, ...
7
votes
0answers
90 views

Complexity and approximability of maximum edge biclique problem on co-comparability graphs

A subgraph $H$ of a given graph $G$ is called a biclique of $G$ if $H$ is a complete bipartite graph. Given a graph $G$, finding a maximum edge biclique is known to be NP-complete (Peeters, Discrete ...
0
votes
1answer
93 views

Planning jobs as partition problem

I think this should be a famous problem but I could not find its name. I have $n$ jobs with size $s_i$ that are offered at time $t_i$ and $k$ machines. How can I assign jobs to machines to minimize ...
2
votes
1answer
251 views

Steiner tree problem for unweighted graphs

Steiner tree problem for weighted graphs is NP-hard. How about unweighted graphs? That is, given a graph $G=(V,E)$ and a subset $C$ of $V$, find a subtree of $G$ with the least number ...