Questions about approximation algorithms.

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Is MAX-SAT SETH (like) hard?

If we make a random assignment to the variables in $k$-sat ($m$ clauses), we are going to satisfy $(1-2^{-k})m$ clauses in expectation. In general satisfying fewer clauses is considered easy. There ...
7
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1answer
210 views

Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning?

I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics). I am particularly ...
3
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1answer
90 views

Set cover in which some pairs of sets are forbidden

I'm trying to find an approximation algorithm for a variant of the weighted set cover problem. However, this variation doesn't seem to let me apply the traditional set cover arguments for an ...
3
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0answers
45 views

Multiple knapsack fpt?

It was an open question whether multiple knapsack is fpt wrt standard parameter. Since at SODA 2009 Jansen has presented an EPTAS for multiple knapsack and an EPTAS implies the existense of an fpt ...
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48 views

Makespan minimization on uniform parallel machines or unreleated machines with different releases dates in polynomial time in machines and jobs

Is there an algorithm for the problem: uniform parallel machines|different releases dates| min makespan ($Q|r_i|C_{max}$) or even unrelated machines|different releases dates|min makespan ...
2
votes
1answer
128 views

Algorithms for computing the minimal vertex separator of a graph

Background: Let $u,v$ be two vertices of an undirected graph $G=(V,E)$. A vertex set $S\subseteq V$ is a $u,v$-separator if $u$ and $v$ belong to different connected components of $G−S$. If no proper ...
0
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1answer
218 views

NP-complete problems with optimal approximation in poly-time

I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not ...
0
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33 views

Understanding Frieze's log(n) approximation algorithm for ATSP

In the Frieze's log$_{2}$(n) approximation algorithm for ATSP, let's consider any $G_{o}$ $\subset$ G, where G is the underlying complete digraph with n vertices, CYC(G$_{o}$) be the cost of optimal ...
2
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59 views

$\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ problems

All the problems which are either $\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ cannot admit a $\mathsf{PTAS}$, unless $P=NP$. I would like to know whether $\mathsf{MaxSNP}$ ...
0
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0answers
98 views

Steiner Tree and minimum spanning tree

If I must connect: $$2^k$$ terminals in a Steiner Tree choosen randomly and connect them with the cheapest component; "loss - contracting algorithm" is a good way? Or is an "Iterative Randomized ...
0
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0answers
53 views

Approximation Algorithm for TSP-like problem

Suppose we are given a graph with distances for each of the edges and merit for each of the nodes. What are the best (approximation) algorithms for computing the the most meritorious simple path with ...
2
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0answers
64 views

Recent insights on algorithms for 1D bin packing

This is just a general question on recent algorithms for the 1D bin packing problem. I just want to collect some information on this issue, so I’m grateful for any information. Especially heuristics ...
2
votes
1answer
252 views

Modifying Hopcroft-Karp algorithm to get approximate bipartite matching

I am trying to find an algorithm to find an $\epsilon$-approximate maximum matching $M_{\epsilon}$ in a bipartite graph in $O(m/\epsilon)$. The partite groups are of equal size, they are $A$ and $B$. ...
5
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0answers
80 views

The distribution on the solution space induced by randomized rounding

Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if $maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut $S$ of fractional size at least ...
0
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0answers
61 views

k closest points that belong to a set

This is a question from theory community, but I came across this issue in a practical problem. So just have this in mind. I have a set of real vectors: $$ S = \lbrace v_1, \dots, v_n \rbrace $$ ...
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67 views

Approximating a max-cut's intersection with other cuts

For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set $S$ and its complement. A max cut is one with at least as many edges as any ...
9
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0answers
105 views

Advances towards proving the Held-Karp conjecture for TSP

I've only began my research into the Held-Karp conjecture and I was wondering about recent progress in proving the conjecture. The Held-Karp relaxation is conjectured to have an integrality gap of ...
3
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1answer
89 views

Approximation algorithms for min vector subset-sum over GF(2)

In this question vzn asked about the following problem, which I'll call Vector-Subset-Sum. Given a set of vectors $v_i$ over GF(2) and a target vector $y$, is there a subset of the $v_i$ summing ...
3
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0answers
57 views

Maximizing a submodular function with restricted values

Maximizing a general monotone submodular function $f$ under the constraint that $|S|\leq k$, can be approximated to $(1-1/e)$. I am wondering if a better approximation algorithm exists if the ...
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0answers
411 views

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 ...
12
votes
2answers
374 views

What is this variant of set cover problem known as?

Input is a universe $U$ and a family of subsets of $U$, say, ${\cal F} \subseteq 2^U$. We assume that the subsets in ${\cal F}$ can cover $U$, i.e., $\bigcup_{E\in {\cal F}}E=U$. An incremental ...
4
votes
2answers
126 views

Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup ...
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206 views

Tuning Parameters of Locality Sensitive Hashing

We have given a set of $n$ binary vectors each of dimension $d$, i.e. a binary matrix of $d*n$. Our goal is to group vectors which are almost similar, $\forall v_i, v_j\in\{0,1\}^d$, we say $v_i$ ...
0
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49 views

Minimum vector sets span spaces cover problem

Instance: a set of vectors $V=\{\mathbf{v}_1,\mathbf{v}_2,...,\mathbf{v}_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$, each of which contains multiple vectors ($V_i$ may not be a subset of $V$). In our ...
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192 views

Can we approximate the number of words accepted by an NFA?

Let $M$ be an acyclic NFA. Since $M$ is acyclic, $L(M)$ is finite. In a related question, it was suggested that exact counting of the number of words accepted by $M$ is $\#P$-Complete. The second ...
2
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1answer
82 views

Approximating the value of k in $k$-mean clustering problem

Consider a set of $n$ points in $R^d$ which are covered by some finitely many (say $k$) unit balls. Can we approximate the value of $k$ by querying only sublinear many points. More precisely, by ...
4
votes
1answer
117 views

Inapproximability of $(\alpha, \beta)$ bi-criteria approximation

An $(\alpha, \beta)$ bi-criteria approximation algorithm for $k$-center is defined as an algorithm that returns a solution whose value is $\beta \cdot OPT$ ($OPT$ being the optimal solution for the ...
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144 views

Clustering in sublinear time/query

Given a set of $n$ points in $R^d$, the goal is to cover them with (finitely many) unit balls such that following conditions satisfy: 1) Minimizing the number of balls that are required to cover all ...
-1
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1answer
53 views

Combinatorial algorithm for load balancing

I have a problem that can be solved with linear programming, but I'm hoping there's a combinatorial algorithm for this (even approximation is fine). This is basically a load balancing problem using ...
0
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0answers
38 views

Estimating Graph/ Network Accuracy

If I'm creating a (social) network using some automatic system, which I know is not 100% accurate but for which I can estimate the rate of error, what, if anything, can I say about the accuracy of the ...
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0answers
43 views

inapproximability of logarithic factor of indepence set [closed]

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
3
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39 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 ...
7
votes
1answer
214 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 ...
2
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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 ...
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0answers
46 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
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0answers
193 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
114 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 ...
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0answers
63 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
158 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
180 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 ...
2
votes
3answers
157 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 ...
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127 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
121 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?
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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 ...
8
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1answer
188 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 ...
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0answers
32 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
97 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
349 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
317 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
75 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 ...