Questions tagged [optimization]
general questions about selecting a best element from some set of available alternatives.
434
questions
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+50
Solving Grouped Weighted Job Scheduling with Release Times and Deadlines on a Single Machine with Multiple Availability Intervals
I'm working on a scheduling problem where I need to schedule a set of n weighted jobs that are partitioned into m groups, where ...
-1
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0
answers
18
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Capacitated Vehicle Routing- Help in understanding a proof
The paper "A Capacitated Vehicle Routing Problem on a Tree" (https://link.springer.com/content/pdf/10.1007/3-540-49381-6_42.pdf) by Shinya Hamaguchi1 and Naoki Katoh stated in the ...
- 1
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64
views
Where does combinatorial optimization beat machine learning algorithms?
For some variant of the Vehicle Routing Problem my algorithm that is based on combinatorial optimization performs a lot better than the algorithms based on machine learning of my competitors. So I ask ...
- 77
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votes
1
answer
74
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An inequality about median of points in higher dimensions
Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\...
- 1
3
votes
1
answer
178
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Can the ellipsoid method be used with a randomized separation oracle?
Suppose we are trying to solve the following optimization problem:
$$
\text{maximize } ~~ c\cdot y
\\
\text{subject to } ~~ y\in S
$$
where the region $S$ is described by an exponential number of ...
- 41
2
votes
0
answers
49
views
Submodular welfare maximization: what is the best known approximation ratio of a deterministic algorithm?
In the submodular welfare maximization problem, there is a set $M$ of items that should be partitioned among $n$ agents. Each agent $i$ has a value function $v_i: 2^M\to \mathbb{R}_+$. All value ...
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90
views
How to prove that a given class of convex programs cannot be solved by linear programming?
Given the following program, where $f, g$ are convex functions:
$$
\text{minimize}~~ f(x)
\\
\text{subject to}~~ g(x)\leq 0
$$
the problem can be solved by convex programming algorithms, but it would ...
- 1,880
0
votes
0
answers
59
views
When does the 'Overlap Gap' property not hold?
The overlap gap property describes problems where good solutions form clusters, so that 2 good solutions are either very close or very far. Problems having this property are problematic for ...
0
votes
0
answers
28
views
Computational complexity of CVaR calculation
I am currently looking for literature discussing the computational complexity of CVaR calculation. At this point the only work I have found is the following.
Mavronicolas, Marios, and Burkhard Monien. ...
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3
votes
1
answer
176
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Solving linear programs with special structure
We have an application and at some point we need to solve a linear programming problem that looks like this:
$$
\min\ w_{1,2} + w_{3,4} + w_{5,6}\\
x_i - x_j \leq c_{ij},\ \forall\ (i,j) \in C\\
x_1 - ...
- 73
3
votes
2
answers
217
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What is this graph problem, and how hard is it?
My problem is quite simple to state, so it surely must have a name:
Given a graph $G=(V,E)$ with edge weights $w(e) \in \mathbb{Z}$, find a $V' \subseteq V$ that maximizes $\sum_{e \in E' } w(e)$, ...
- 31
1
vote
1
answer
75
views
A bound that follows from submodularity
I am studying Lemma 1 of this paper: The Adaptive Complexity of Maximizing a Submodular Function. The proof appears on page 11.
I got stuck on this inequality:
where $f$ is a monotone submodular set ...
- 113
1
vote
0
answers
88
views
How to solve the following continuous optimization problem?
Consider a function $f: X\times Y\times N$, where $X, Y \subseteq \mathbb{R}^m$ are convex sets, and $N = \{1,2,\dots,n\}$. We additionally know that
$f(\cdot,y,S)$ is convex for fixed $y,S$
$f(x,\...
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1
vote
0
answers
57
views
Unbounded Knapsack Instance with a Single Optimum that takes each Item Once?
Consider the Unbounded Knapsack Problem (UKP): We are given a set of $n$ items $I = \{1,\ldots,n\}$ of integral weights $w_1, \ldots, w_n \in \mathbb{N}$, integral profits $p_1, \ldots, p_n \in \...
- 73
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55
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Uniformly redistributing items across bins. What problem is this?
I'm trying to find reading material on a particular problem I'm interested in, but I don't know the terms to search.
Problem assumptions/definitions:
We have finite number of items I with weights [0, ...
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1
vote
0
answers
44
views
Will the maximum entropy joint distribution given a known set of marginal distributions have the maximum plausible support?
Define $[n] = \{1, 2, ..., n\}$. Given a distribution $P : \{0, 1\}^{[n]} \rightarrow [0, 1]$ and a subset $S \subseteq [n]$, we can define the $S$-marginal of $P$, $P_S : \{0, 1\}^S \rightarrow [0, 1]...
- 1,562
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0
answers
17
views
Why training a GAN with simultaneous optimization of the loss functions does not ensure good images?
I am training a GAN in which I am optimizing or reducing the loss of the generator and the discriminator simultaneously. However, the images generated are very noisy. What could be the hidden reason? ...
- 101
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74
views
Packing k vertex trees
Consider a graph $G=(V,E)$ with $n$ vertices.
What do we know about packing of $k$ vertex trees, Both integral and fractional packing are interesting.
$k=2$, it is just the number of edges, hence ...
- 4,327
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votes
0
answers
26
views
Quantifying the cost of procedures
Is there any research on quantifying the cost of a procedure, with regard to compiler optimization?
I.e. assigning some kind of cost in terms of CPU time or memory to a procedure, either so the ...
- 111
3
votes
0
answers
73
views
Solving MDPs with polytope action spaces
A (finite) Markov Decision Process (MDP) consists of a finite set of states $S$, a finite set of actions $A_s$ which we will allow to depend on the state $s\in S$, an initial state $s_0\in S$ (the ...
- 2,151
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109
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In the Schönhage-Strassen algorithm for integer multiplication, when we calculate the product of two n-bit integers, why do we do so modulo 2^n + 1?
I ask this because it seems to me that there might be a loss of information here. The product of two n-bit integers could be up to 2n bits long, but any element of the integers modulo 2^n + 1 is at ...
1
vote
0
answers
54
views
Find the minimum cost spider joining a root to some leaves
A spider is a tree with at most one vertex of degree greater than 2. This vertex is called the head of the spider.
I am interested in the following problem: We are given an undirected graph $G = (V,E)$...
- 597
1
vote
0
answers
58
views
The tree augmentation problem, but with hyperlinks
In the (Weighted) Tree Augmentation Problem, we are given a tree $T = (V,E)$ and a set of additional edges $L$ called links with non-negative costs. Each link $\ell = (u,v)$ covers the tree edges ...
- 597
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0
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25
views
Custom data structure for subarray problems
Is it possible to build a data structure for solving subarray related problems efficiently (E.g. counting the number of subarrays of an array satisfying a given condition)?
4
votes
1
answer
124
views
Tensor network contraction "bubbling": why are some approaches more computationally efficient than others (question from a beginner)
I am learning the very basics of tensor network theory and I am trying to understand why some ways to contract tensors are better than others in terms of computational complexity. Knowing in which ...
1
vote
2
answers
96
views
Growth rate of Knapsack Solutions
Let's say I have a Knapsack with capacity $\tau$, and I have an infinite sequence of items with weights $(a_n)_{n=1}^{\infty}$. A feasible Knapsack solution is a subset $S \subset \mathbb{N}$ such ...
- 607
2
votes
1
answer
195
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Maximize a special monotone submodular function - is it easier?
I am looking for a way to optimize the function $f$, defined below.
First, fix some positive integer $k$ and let $c_1$ and $c_2$ be non-negative vectors in $\mathbb{R}^n$. Let $g$ be an increasing ...
- 597
0
votes
0
answers
70
views
Parameterized Complexity of Vertex Multicut
Let $G$ be an undirected graph, $\{(s_1,t_1),\dots,(s_k,t_k)\}$ a collection of pairs of vertices, and $p$ an integer. The Vertex Multicut problem asks if there is a set $S$ of at most $p$ vertices ...
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votes
0
answers
102
views
Optimization problems where the solver can choose which variables are continuous
A typical optimization problem looks like the following, where $f$ represents the objective and $g$ the constraints:
$$
\text{maximize}~~~f(x1,\ldots,x_n)~~~\text{subject to}:
\\
g(x_1,\ldots,x_n)=0,
\...
- 1,880
2
votes
1
answer
131
views
Is there an approximate version of the strong duality theorem for linear programming?
Consider the following dual linear programs:
$$
\min \mathbf{c^T x} ~~ \text{s.t.} ~~ A \mathbf{x} \geq \mathbf{b}, \mathbf{x}\geq 0;
\\
\max \mathbf{b^T y} ~~ \text{s.t.} ~~ A^T \mathbf{y} \leq \...
- 1,880
2
votes
0
answers
85
views
Does the standard 4/3 integrality gap for TSP example work for Euclidean TSP?
Given a graph $G=(V,E)$, costs $c \in \mathbb{R}^E$ the TSP problem is to compute a min cost tour of the graph. The LP is
min $ c^tx $
$x(\delta(S)) \geq 2 \ \ \ \ \forall S \subset V $
$x(\delta(v)...
- 227
2
votes
0
answers
94
views
Can this relaxed subset-sum problem be solved with a smaller dynamic program? [closed]
Cross-post from CS.SE
In the subset sum problem, the input is a list of positive integers $x_1,\ldots,x_n$ and an integer $T$, and the goal is to decide whether there is a subset of sum exactly $T$.
...
- 1,880
4
votes
0
answers
96
views
Analogue of Chow-Liu tree for $L_1$
Say $\Omega$ is a finite set and $f$ a probability mass function (pmf) over $\Omega^d$. Now let $T$ be a spanning tree on the set $V=\{1,2,\ldots,d\}$, and consider a collection of one- and two- ...
- 10.2k
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0
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110
views
optimization on graph edges selection
I have the below problem. I wonder if there exists a similar known class of problems (e.g., in optimization, graph theory) which I can relate my problem to, and find a similar solution there.
I am ...
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1
vote
1
answer
130
views
3-SAT runtime if an optimal order to eliminate possible solutions is known
As a mental exercise I have been playing around with the 3-SAT problem, but I am having difficulty proving or disproving the usefulness of a current idea that I am playing around with.
My current ...
8
votes
0
answers
148
views
What is the least compressible probability distribution? (under entropy constraint, for an expected squared error metric)
Consider a distribution $\mathcal D$ over the reals, a real parameter $H\in\mathbb R^+$, and an integer parameter $k\in\mathbb N$.
The Entropy-Constrained Quantization problem asks what is the best ...
- 9,378
3
votes
0
answers
135
views
Bin packing where each item must occur in $k$ bins
I am looking for information on a generalization of bin-packing in which each item should appear in exactly $k$ different bins, for some positive integer $k$. The standard bin packing problem ...
- 1,880
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votes
0
answers
48
views
Is there a primal-dual algorithm for the Tree Augmentation Problem or the Cactus Augmentation Problem?
The TAP problem and the CacAP problem can be seen as covering problems for the minimum cuts of a graph.
It seems like these problems would fall under the framework of network design problems (...
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1
vote
1
answer
61
views
Maximize the absolute value of connected nodes after $k$ modifications
Given a graph $G=\{V,E\}$, each node $i$ has a value $v_i$. Given budget $k$, we have $k$ chance to add 1 or minus 1 for a node's value, for example, $v'_i=v_i+1$ or $v'_i=v_i-1$. In particular, $v'_i$...
- 57
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votes
0
answers
52
views
Finding the best $k-$subset which maximizes a matrix sum
Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem:
Is there a polynomial-time solution to the following ...
1
vote
0
answers
32
views
Reference showing global optimality of local minima for matrix factorization
Consider the following matrix factorization problem: Given an $n\times m$ matrix M, find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $||UV^T - M||_F^2$ is minimized.
I have heard it ...
- 111
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votes
0
answers
50
views
Using bin-packing algorithms to approximate maximum-makespan
Bin-packing (BP) and maximum-makespan (MM) are dual problems. In both problems, the input can be defined as a set $S$ of positive integers, and the output is a partition of $S$.
In BP, there is a ...
- 1,880
1
vote
1
answer
128
views
Partition the edges of a bipartite graph into perfect $b$-matchings
Any $r$-regular bipartite graph can be partitioned into $r$ disjoint perfect matchings.
I want to know whether a version of this extends to perfect $b$-matchings.
Suppose we have a bipartite graph $G =...
- 597
0
votes
1
answer
187
views
Does an upper bound on the integrality gap imply an approximation algorithm with the same ratio?
Often, we can model combinatorial optimization problems with an Integer Program. Then there is an associated Linear Relaxation which drops the integrality constraints on the variables.
Let's say we ...
- 597
0
votes
0
answers
99
views
Examples of SDP constant approximation algorithms on minimisation problems
I was recently going through a survey on semidefinite programming and its use in approximation algorithms. Here are some problems I am familiar with that have SDP approximations:
Max Cut ($\approx 0....
- 862
1
vote
0
answers
42
views
Modifying sets to minimize the distance among each pair of the mean value of sets
Given $n$ points, each point $x_i$ has a value $v_i \in \mathbb{R}^{d}$, and there are $m$ point sets $\{S_1,\dots, S_m\}$ that each point set consists of some points. The size of point sets can be ...
- 57
7
votes
1
answer
288
views
Hardness of maximizing $x^TAy$ with $\{-1,1\}$ entries
My question concerns the NP-hardness of the following discrete optimization problem:
Given a matrix $A \in \{ \pm 1 \}^{m\times n}$,
$$\begin{array}{ll} \underset{x \in \{ \pm 1 \}^m ,\, y \in \{ \pm ...
- 73
-2
votes
1
answer
318
views
Find research partner (profession and beginner)
I've 10 years of industrial work, but in my free time, I do research, write papers to conferences, help to teach to my old friend at the university and I even did a Ph.D. full-time program.
Now, I've ...
- 45
1
vote
2
answers
219
views
Finding the point that maximizes a linear function
Consider $N$ two-dimensional points of the form $(x_i, y_i)$ where all $x_i, y_i > 0$ are positive integers. We will be given a workload of queries $Q = \{c_1, \dots, c_k\}$ where for each $c_j \in ...
- 1,167
3
votes
1
answer
157
views
TSP with "enemy" nodes
I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for.
The story/idea ...
- 407