Questions about approximation algorithms.

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2
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1answer
70 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?
4
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0answers
76 views

Barrier Coverage for a Polygonal Area in the Plane

Given a region $R$ in the plane, the so called Barrier Coverage problem asks to place sensors in such a way that any intruder going (continuously) into $R$ is detected. This problem received quite ...
5
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0answers
102 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 ...
4
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1answer
110 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
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0answers
25 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
58 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
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1answer
302 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
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2answers
115 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
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1answer
54 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 ...
3
votes
1answer
85 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
86 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
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2answers
180 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
122 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
68 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
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1answer
54 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
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2answers
289 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
114 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
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1answer
228 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
157 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
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0answers
174 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
103 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)?
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0answers
49 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
80 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 ...
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0answers
52 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
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0answers
56 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 ...
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0answers
42 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
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0answers
62 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
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1answer
222 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 ...
2
votes
2answers
144 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
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1answer
106 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
75 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 ...
9
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0answers
112 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
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1answer
87 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
431 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
72 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
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2answers
242 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 ...
3
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0answers
151 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
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0answers
83 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
89 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
197 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 ...
2
votes
1answer
145 views

Approximate Maximum Weight Matching

I am looking for an approximated (or randomized) maximum weight matching algorithm. Do you have any suggestion for me? In my problem, I have a bipartite graph with N abound 1000 (#vertices on each ...
5
votes
1answer
135 views

Best approximation for a HYPERGRAPH-MAXDICUT problem

Consider a $(c^a,(c+d)^a,1)$-regular directed hypergraph $\mathcal{H}(a)$ on $n^a$ vertices with fixed $n\geq c+d+1$, fixed $c\geq 2$, fixed $d\geq 0$ and variable parameter $a\geq 1$ (meaning every ...
5
votes
1answer
111 views

PTAS for a variant of multiple knapsack?

I am interested in the following variant of multiple knapsack. There are $n$ items with values $v_1,\ldots,v_n$. We want to partition the items into $m$ subsets $S_1,\ldots,S_m$ to maximize ...
2
votes
0answers
38 views

Approximating BLEDP on restricted graph classes

In the edge-disjoint paths (EDP) problem, we are given an undirected graph $G$, and a set $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$ of $k$ source-sink pairs. The objective is to maximize the number of ...
2
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0answers
56 views

Approximation algorithms for multicut for special classes of graphs

The multicut problem is the following. Given a graph $G=(V,E)$ with edge costs $c_e$ for edge $e$ and a set of $k$ terminal pairs $\{(s_1,t_1),\ldots,(s_k,t_k)\}$, the objective is to find a set of ...
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1answer
88 views

what problem is this? [closed]

I have this instance: Let's say I have two (could be more) friends, one weighing 200 pounds and another weighing 100 pounds; I won a box with 30 chocolates in a contest and I want to divide among ...
2
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0answers
96 views

The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...
0
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0answers
93 views

Finding assignment-minimum complete k-partite graph cover

Is there any work on approximation algorithms (or exact algorithms) for finding an assignment-minimum cover of an arbitrary graph using complete k-partite subgraphs? I'm assuming this problem is ...
1
vote
2answers
126 views

algorithms to split data into roughly equal sized quantiles

What is the state-of-the art on algorithms that calculate/estimate approximate quantiles? I don't even worry about errors in terms of the value of quantiles (here meaning the cutoff) but having ...
13
votes
1answer
202 views

Good reference about approximate methods for solving logic problems

It is known that many logic problems (e.g. satisfiability problems of several modal logics) are not decidable. There are also many undecidable problems in algorithm theory, e.g. in combinatorial ...