Questions tagged [optimization]

general questions about selecting a best element from some set of available alternatives.

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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 ...
0 votes
0 answers
63 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 ...
0 votes
1 answer
74 views

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 $\...
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3 votes
1 answer
178 views

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 ...
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 ...
0 votes
0 answers
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 ...
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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. ...
3 votes
1 answer
176 views

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 - ...
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3 votes
2 answers
217 views

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 $\sum_{e \in E' } w(e)$, ...
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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 ...
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,\...
1 vote
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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 \...
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55 views

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
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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]...
0 votes
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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? ...
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1 vote
0 answers
72 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 ...
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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 ...
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 ...
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109 views

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 $G = (V,E)$...
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 ...
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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
123 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 ...
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2 votes
1 answer
195 views

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 ...
0 votes
0 answers
69 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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2 votes
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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, \...
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 \...
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)...
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2 votes
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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 $T$. ...
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- ...
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0 votes
0 answers
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 ...
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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 ...
0 votes
0 answers
47 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 (...
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, $v'_i$...
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0 votes
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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 ...
0 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 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 =...
0 votes
1 answer
185 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 ...
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....
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 ...
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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 ...
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-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 ...
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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 ...
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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 ...
2 votes
1 answer
202 views

Characterization of integral polyhedra

A rational polyhedron $P \subseteq \mathbb{R}^n$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $P = conv(P \cap \mathbb{Z}^n)$. Equivalently, $P$ is integral if ...

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