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Questions tagged [approximation-algorithms]

Questions about approximation algorithms.

22
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2answers
13k views

Universal Approximation Theorem — Neural Networks

I posted this earlier on MSE, but it was suggested that here may be a better place to ask. Universal approximation theorem states that "the standard multilayer feed-forward network with a single ...
44
votes
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 ...
48
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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 ...
32
votes
1answer
1k views

Toy Examples for Plotkin-Shmoys-Tardos and Arora-Kale solvers

I would like to understand how the Arora-Kale SDP solver approximates the Goemans-Williamson relaxation in nearly linear time, how the Plotkin-Shmoys-Tardos solver approximates fractional "packing" ...
35
votes
3answers
3k views

Max-cut with negative weight edges

Let $G = (V, E, w)$ be a graph with weight function $w:E\rightarrow \mathbb{R}$. The max-cut problem is to find: $$\arg\max_{S \subset V} \sum_{(u,v) \in E : u \in S, v \not \in S}w(u,v)$$ If the ...
16
votes
2answers
701 views

Set Cover for Permutation Matrices

Given a set S of nxn permutation matrices (which is only a small fraction of the n! possible permutation matrices), how can we find minimal-size subsets T of S such that adding the matrices of T has ...
5
votes
1answer
689 views

Maximizing a monotone supermodular function s.t. cardinality

I've tried to comb the literature and seen a lot of references to results that almost but don't quite seem to address this. Question: Is it known to be true or is there a hardness result ...
44
votes
8answers
5k views

The importance of Integrality Gap

I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real ...
18
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3answers
1k views

Integrality gap and approximation ratio

When we consider an approximation algorithm for a minimization problem, the integrality gap of an IP formulation for this problem gives a lower bound of an approximation ratio for certain class of ...
26
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4answers
1k views

Compendium of the Best Approximation and Hardness Results for NP optimization problems

Do you know any up-to-date wiki dedicated to NP optimization problems with their best approximation and hardness result? Based on the feedback, it seems that it is safe to assume there is not such a ...
21
votes
2answers
1k views

Polynomial time approximation algorithms for machine scheduling: how many open problems are left?

In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper "Polynomial time approximation algorithms for machine scheduling: Ten open problems". Since then, to the best of my knowledge, ...
24
votes
4answers
2k views

Bounded-cardinality bounded-frequency set cover: hardness of approximation

Consider the minimum set cover problem with the following restrictions: each set contains at most $k$ elements and each element of the universe occurs in at most $f$ sets. Example: the case $k = 4$ ...
16
votes
0answers
410 views

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 ...
13
votes
1answer
333 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 ...
11
votes
5answers
6k views

Is there any gradient descent based technique for searching absolute minimum (maximum) of a function in multidimensional space?

I'm familiar with gradient descent algorithm which can find local minimum (maximum) of a given function. Is there any modification of gradient descent which allows to find absolute minimum (maximum),...
12
votes
3answers
3k views

Is there an online-algorithm to keep track of components in a changing undirected graph?

Problem I have an undirected graph (with multi-edges), which will change over time, nodes and edges may be inserted and deleted. On each modification of the graph, I have to update the connected ...
11
votes
2answers
3k views

Any fast algorithm for minimum cost feedback arc set problem?

In a directed graph, $G=(V,E)$, $F\subset E$, if $G\setminus F$ is a DAG(directed acyclic graph), $F$ is called a feedback arc set. If each edge is associated with a weight $w$, the minimum cost ...
12
votes
3answers
625 views

A Multi-cut Problem

I'm looking for a name or any references to this problem. Given a weighted graph $G = (V, E, w)$ find a partition of the vertices into up to $n = |V|$ sets $S_1,\ldots,S_n$ so as to maximize the ...
5
votes
3answers
431 views

Partitioning a segmented stick

Problem : We are given a stick partitioned into n - equal parts. Each of these parts has a weight, let's say x. Number of times x appears as weight of some part is guaranteed to be even. For ...
1
vote
1answer
322 views

Questions about computing matrix rigidity

Matrix rigidity was introduced by Valiant in 1977: The rigidity $Rig_M(r)$ of boolean matrix $M$ over GF(2) is the smallest number of entries of $M$ that must be changed in order to reduce its ...
7
votes
1answer
677 views

Approximating transitive reduction of a transitive closure of a dag

Let's suppose a transitive closure $G^+$ of a dag $G$ is given and we want to compute an approximation of the transitive reduction $G^-$ such that the full transitive reduction is a subgraph of the ...
7
votes
3answers
2k views

A simple approximation algorithm for the TSP

Consider the following extremely simple approximation algorithm for the TSP. Input: A complete weighted graph $G=(V,E).$ Take any three vertices $a,b,c\in V$ and let $H:=(a,b,c,a).$ While there ...
4
votes
3answers
469 views

bin packing with overlapping objects

I have $N$ bins with capacity $M$ and $k$ objects with size $s_i$. The goal is to pack these objects in the bins. Until now it is similar to the bin-packing problem. But the twist is that each object ...
9
votes
0answers
248 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 ...
8
votes
1answer
892 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 P4-...
7
votes
2answers
666 views

Approximation algorithms for MAX-CUT, when sizes of partition sets are fixed

The MAX-CUT problem has constant factor approximation, but we can't control the sizes of the sets in resulting partition. What is known about maximizing cut size, if we restrict one part of the ...
6
votes
1answer
292 views

Approximation schemes for P-complete problems?

What work has been done on approximation schemes for $\mathsf{P}$-complete optimization problems? Would the desired approximation algorithms here be "fully log-space approximation schemes" or "fully $\...
5
votes
2answers
348 views

A variant of maximum matching: disjunctive constraints on the endpoints' degrees of edges in matching

The question is asked first at here. It described what the problem is and a trival greedy algorithm. Also the accepted answer gave a proof of its NP-completeness. Problem: Given a graph $G(V,E)$, ...
4
votes
0answers
110 views

Approximating Front Size of Asymmetric Matrices

The front size of a matrix $A$ is the largest number of non-zeros below the diagonal in any column of its Cholesky factor. If $A$ is symmetric then the minimum front size of $A$ is equal to the ...
3
votes
0answers
132 views

Exactly solvable but non-trivial integrality gap

Are there interesting polynomial time solvable problems that we know of for which the natural convex relaxation has a non-trivial integrality gap? Note: Maximum matching doesn't qualify because I ...
2
votes
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
0answers
205 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 ...
1
vote
1answer
422 views

The Goemans-Williamson algorithm in the $SOS$ framework

If there is a variable $x_i$ for every vertex $i$ of a $d$-regular graph $G$ then assigning $x_i = \pm 1$ gives a cut, say $(S,\bar{S})$, of the graph. We can then see that, $\langle x,L x\rangle$, ...