# Questions tagged [approximation-algorithms]

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### What is the communication complexity of approximating addition?

In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. Specifically, the class of problems I am interested in is parametrized by four ...
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### finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
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### Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
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### Example of a hardness-of-approximation proof which improves the approximation factor?

Are there any examples of hardness-of-approximation reductions where we get a better hardness bound for the problem we've reduced to than the problem we've reduced from? In the examples I've seen so ...
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### Count satisfying assignments of CNF formulas over all possible negation assignments

Consider the set of all CNF instances that can be generated by adding negations to a single monotone CNF instance. How hard is it to compute the sum of the counts of satisfying assignments for the set?...
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### Machine Learning Algorithm To Fill Data Holes

I'm having trouble finding a good place to begin with this. I'm just looking for a name or point to start researching a Let's say I have 1000 records. 10 of these records are only 90% complete. The ...
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### Approximation for counting the number of simple $s$-$t$ paths in a general graph

I have been told that there are some good polynomial time algorithms for approximating the number of simple paths in an directed graph from given starting vertex $s$ to given ending vertex $t$. Does ...
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### 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 ...
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### 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 ...
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### 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 $n_0$, ...
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### Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour  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 ...
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### 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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### 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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### 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 / $k$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $W$. ...
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### 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 valid ...
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### 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 ...
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### 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?
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### 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 ...
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 ...