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Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

3
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1answer
121 views

Intuitive explanation behind Goemans-Williamson randomized rounding

A very simple randomized cut algorithm achieves $1/2$ of the optimal value: just choose each vertex to be in the cut with probability $1/2$, independently. Goemans-Williamson does something more ...
0
votes
2answers
805 views

Is the current best approximation ratio for Vertex Cover problem also a lower bound?

In textbook "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein. in pp.1110-1111, they argue that the vertex-cover problem is a 2-approximation algorithm and it is lower bound so we ...
-1
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0answers
55 views

greedy performance

I have a set function $f:2^V\rightarrow R_+ $which is non negative monotone supermodular function with a property that $f(\{x\})$ is same for all $x\in V$($f(\{x_1\})=f(\{x_2\})=\dots =f(\{x_i\}),\...
7
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1answer
106 views

Smoothed analysis to compare algorithms

Has there been any research using smoothed analysis to compare approximation algorithms that have the same approximation ratio? Any research that compares algorithms using smoothed analysis would be ...
5
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1answer
311 views

Do combinatorial discrepancy upper bounds lead to smaller $\epsilon$-nets (as with $\epsilon$-samples)?

An $\epsilon$-sample (or $\epsilon$-approximation) of a family of subsets $\mathcal{S}$ of a ground set $X$ is a subset $P \subseteq X$ which preserves relative sizes of sets up to $\epsilon$. I.e., ...
44
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4answers
12k views

Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
2
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0answers
65 views

What is the competitive ratio of a $d$-way associative LRU cache?

In a caching problem, items arrive online, and the algorithm needs to decide which elements to keep in the cache. If the current item is not cached, we pay a penalty of $1$. It is well known that for ...
2
votes
1answer
90 views

Maximum-minimum satisfiability

In MAX-SAT, given a formula, we want to maximize the number of satisfied clauses: given a formula $\phi = c_1 \cap \cdots \cap c_n$, where each $c_i$ is a disjunction, we want to find the largest $k\...
3
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2answers
108 views

Minimum relevant variables in linear system - additive approximation

In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.: $$ A x = b $$ and the goal is to find a solution $x$ with as few nonzero variables as ...
17
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3answers
435 views

Is there a constant factor approximation algorithm for 2D rectangle coloring problem?

The problem we consider here is the extension of the well-known interval coloring problem. Instead of intervals we consider rectangles having sides parallel to axes. The objective is to color the ...
2
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0answers
153 views

Best polynomial-time approximation factor for NP-optimization problems

Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold: There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
6
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0answers
110 views

Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour [1] states that if graph G has large treewidth, then it contains a large grid as minor. Most approximation results on general classes of graphs with ...
2
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1answer
113 views

About a pre-processing step for primal–dual weighted set cover problem

I was reading the paper titled "Primal-dual RNC approximation algorithms.." by Rajagopalan and Vazirani. I have a problem of understanding the Lemma 4.1.1. They present a dual fitting based algorithm ...
1
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0answers
33 views

PTAS for projective clustering : survey

$(k,j)$-projective clustering is the natural generalisation for k-clustering, in which one needs to find $k$ $j$-flats in $\mathbb{R}^d$ that minimizes the cost function as defined below: Given a $j$-...
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0answers
35 views

Best approach for allocation problem

I am a bit rusty on optimization algorithms and need an advice. This is my problem: I have n images (with width and ...
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0answers
49 views

Why do k min-hashes, instead of one hash where we find the k minimum elements?

Traditionally if one wants to sketch streams for Jaccard similarity hashing, one finds the minimum element in each of $k$ permutation for comparison purposes, and then takes number_of_collisions / $k$ ...
2
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1answer
101 views
6
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1answer
168 views

Does Max Planar 3-SAT admit a PTAS?

Suppose we are given a formula $\phi$ of 3-SAT, with variables $x_1,\dots, x_n$ and clauses $C_1,\dots, C_m$. Consider the graph $G_\phi$ where there is one node for each clause $C_i$, for each ...
5
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0answers
143 views

Classification of randomized approximation algorithms

Is there a known classification of randomized approximation algorithms, in the same vein as the distinction between Monte Carlo and Las Vegas algorithms for decision problems? (Or equivalently ...
0
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1answer
56 views

Polynomial approximation algorithm for set cover with assumption

We want to cover $n$ elements with some sets from $S_1, …, S_m$ (classical set cover). We furthermore suppose that any element belongs to at least $k$ sets and want to find a set cover with cardinal ...
2
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0answers
85 views

Counting the maximum number of paths of length $n$ that differ in at least $k$ edges

What is known about the complexity of solving (or approximately solving) the following problem? INPUT: Graph $G=(V,E)$ and constants $L$ and $K$. OUTPUT: The maximum size of any set $S$ of simple ...
3
votes
2answers
167 views

Finding a set which dominates the Minimum Dominating Set

Given an unweighted, undirected graph, a dominating set $S$ is a set of nodes such that every node is in $S$ or adjacent to a node in $S$. The dominating set problem is NP-hard, but I am considering ...
6
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1answer
141 views

Brute force search algorithm for semidefinite programming (representation of spectrahedron)

I was wondering if there exists a brute force search algorithm for semidefinite programming problems. Specifically, can we find finite number of points in the positive semidefinite cone such that for ...
2
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0answers
33 views

Approximate answer to Max-SMT (bitvector) query

I have a problem that could be completely solved as Max-SMT instances in the theory of quantifier-free bit-vectors, but it apparently is too complex to be tractable with current Max-SMT technology. ...
4
votes
2answers
137 views

Max cut problem between two connected subgraphs

Let $G$ be a connected graph. Consider the problem of finding a partition $G = A \cup B$ into connected subgraphs, so that the cut between $A$ and $B$ is maximized. Is there anything which is known ...
3
votes
1answer
48 views

Complexity of distributively verifying that the diameter is small

Consider a graph $G=(V,E)$ and an integer parameter $k$. I'm interested in the round complexity, in the CONGEST model, of checking if the diameter of the graph is "much larger" or "much smaller" than ...
1
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0answers
54 views

k-center 2.0: A stronger k-center condition

Given an unweighted, undirected graph, we can use the classical 2-appx for $k$-center to select a set $S$ of centers such that every vertex is within a distance of 2 of some center in $S$. Note that ...
0
votes
1answer
96 views

How can algorithms with nested combinatorial searches be quasi-linear?

I've come across algorithms similar to the one below, where a demanding step is performed, which should have at least polynomial complexity, yet the whole algorithm is deemed quasi-linear without ...
48
votes
3answers
2k views

Is there a sensible notion of an approximation algorithm for an undecidable problem?

Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be ...
6
votes
2answers
671 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 ...
9
votes
4answers
239 views

W[1]-hard problems with FPT time approximation algorithms

I'm looking for problems that are hard to solve in FPT time but has an approximation algorithm. That is, problems that are: R1. W[1]-hard. R2. Admit a (preferably constant) approximation algorithm ...
2
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0answers
66 views

Is there any online problem that the best known competitive ratio is better than the hardness of approximation (or the best known approximation)?

In online-setting, we usually allow exponential time to think and come up with a strategy. On the other hand, in offline-setting, we might care about solving a particular problem optimally, or as good ...
1
vote
1answer
54 views

Does the following 2-rounds distributed algorithm approximates a maximal matching well?

Let $G$ be an undirected graph. I'm looking for a two-rounds distributed algorithm that matches as many vertex pairs as possible. Consider the following protocol for vertex $v$. Use a fair coin to ...
3
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0answers
221 views

Can we find a generic tight example for the greedy algorithm of the unweighted generalized assignment problem?

In the unweighted generalized assignment problem (UGAP) we have $n$ items and $k$ knapsacks. For each item $i$ and knapsack $j$, there is a weight $w_{ij}$. Also, every knapsack has a capacity $W$. ...
0
votes
1answer
98 views

Is there any better than (2/k)-approximation algorithm for Independent Set in Coloring graph?

I would like to know any results in terms of approximation algorithm about Maximum weight Independent Set problem in coloring graph. Input: Given a graph G with non-negative vertex weights and ...
1
vote
1answer
100 views

Minimizing SubModular Function: Cardinality

Given a submodular function f: 2^V to reals (not necessarily monotone), and an integer k, find a set S such that |S| <= k and such that f(S) is minimized. When the size constraint is |S| >=k, the ...
0
votes
1answer
46 views

About using smoothness of the Hessian for getting to approximate criticality of a non-convex objective

Is there any algorithm which shows that under the assumption of Lipschitz smoothness of the Hessian of a non-convex function one can get to its critical point faster?
1
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0answers
58 views

What is the fastest gradient based algorithm to get to criticality of a “nice” non-convex function?

I am allowing for the following properties for a once differentiable non-convex $f : \mathbb{R}^d \rightarrow \mathbb{R}$, (a) Let there be a $\sigma >0$ s.t the norm of the gradient of the ...
2
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0answers
125 views

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 ...
1
vote
0answers
32 views

Getting to local/global minima with (stochastic) gradient descent on non-convex objectives

Is there any class of non-convex objective functions for which (stochastic) gradient descent can provably get to a local or a global minima? (..maybe in the approximate sense like a point such that ...
4
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0answers
38 views

Approximating a monotone submodular function using a concrete coverage function

Is it possible to approximate a monotone submodular function using a concrete coverage function, i.e. Given a ground set $U$ and a monotone submodular function, $f:2^U\to \mathbb{R}$, the goal is to ...
1
vote
1answer
125 views

Monotone supermodular function minimization under partition matroid constraints

Is there a known approximation algorithm for the problem of minimizing a monotone (non increasing) supermodular function under partition matroid constraints ?
5
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0answers
105 views

Learning hidden variable distribution

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 ...
3
votes
0answers
119 views

Stacks serving interval storage requests

An interval storage request is represented by a tuple $(s,t,v)$ satisfying $s<t$, meaning that the value $v$ needs to be stored from time $s$ to time $t$. A stack serves the request $(s,t,v)$ in ...
2
votes
0answers
84 views

Unbalanced connected partition

Let $G = (V, E)$ be a connected graph with (possibly negative) vertex weights $w(v)\in\mathbb{Z}$. We want to partition the vertices into two parts such that the induced graphs $G'$ and $G''$ are ...
14
votes
2answers
430 views

Proof assistant usage in complexity theory research?

Considering the topics covered at a conference like STOC, are any algorithm or complexity researchers actively using COQ or Isabelle? If so, how are they using it in their research? I assume most ...
0
votes
2answers
138 views

Geometric max cover

Consider $n$ points and a distance function $d$ that satisfies the triangle inequality. We are also given a number $r$. Each point $p$ defines a set $B_p$ (or a ball) that covers all other points ...
1
vote
1answer
347 views

Common terminology used for lower/upper bounds

Suppose you have developed an upper bound on the number of vertices of a particular graph. This bound is the best possible bound that can be found for any given instance. What do you call such a bound?...
2
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0answers
107 views

On approximating problems in $\#P$

We know that for every counting problem $\#A$ in $\#P$, there is a probabilistic algorithm $\mathcal C$ that on input $x$, computes with high probability a value $v$ such that $$(1 − ε)\#A(x) ≤ v ≤ (1 ...
5
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0answers
105 views

An optimal subspace projection problem

Suppose we have a $k$-dimensional subspace $V$ in $\mathbb{R}^n$ given by a basis $\{v_1,\cdots,v_k\in \mathbb{R}^n\}$, find an index set $I\subset [n]$ with $|I|=m$ where $k\le m\le n$, such that $$\...