Questions tagged [optimization]

Questions about problems that entail selecting the best element from some set of available alternatives, and methods to solve them.

Filter by
Sorted by
Tagged with
7
votes
1answer
384 views

In s-t directed graph, how to find many small cuts?

Solving the maximum flow problem yields one qualified minimal cut. But I want several (maybe hundreds) small cuts as candidates. The cuts don't have to be minimum cuts, as long as they are small (in ...
7
votes
1answer
4k views

BFGS vs L-BFGS — how different are they really?

I am trying to implement an optimization procedure in Python using BFGS and L-BFGS in Python, and I am getting surprisingly different results in the two cases. L-BFGS converges to the proper minimum ...
7
votes
2answers
105 views

CNF form of variable assignment problem

There are n variables $x_1$, $x_2$,..., $x_n$ and each one of them takes values from 1 to k (k>= n) and all are distinct. How can I represent this in the CNF form? (I tried the trivial way of trying ...
7
votes
1answer
283 views

Are coevolutionary “Free Lunches” really free lunches?

In their paper "Coevolutionary Free Lunches" David Wolpert and William Macready describe a set of exceptions to the No Free Lunch theorems they proved in an earlier paper. The exceptions involve two-...
7
votes
1answer
4k views

Weighted interval scheduling with m-machines

I am looking for some advice and direction on solving the weighted interval scheduling problem with $m$-machines to plan some experimental "wet lab" procedures. The problem is very similar to the ...
7
votes
1answer
168 views

Hardness of a constrained quadratic maximization

Consider the following quadratic maximization: \begin{align} \max_{\mathbf{x} \in \mathcal{X}} &\quad\mathbf{x}^{T}\mathbf{A}\mathbf{x} \end{align} with \begin{align} \mathcal{X} = \lbrace \mathbf{...
7
votes
1answer
2k views

Greedy strategy for computing the minimum number of rays that hit all balloons

The minimum zap problem below is Exercise 11 in Jeff Erickson's lecture on "Greedy Algorithm". The minimum zap problem can be stated more formally as follows. Given a set $C$ of $n$ circles in the ...
7
votes
0answers
192 views

Algorithms for curve construction

I am interested in algorithms that construct continuous curves between two points in such a way that minimizes an energy functional of the curve. What sort of algorithms are most used for such tasks? ...
7
votes
0answers
126 views

Boolean formula that agrees with most truth assignments

Let $X_1,\dots,X_n$ be $n$ boolean variables. I have an unknown predicate $P(X_1,\dots,X_n)$ on these boolean variables. Of course, I can view the predicate as a function $f_P : \{0,1\}^n \to \{0,1\}...
7
votes
0answers
158 views

Overlap Maximization problem

Here's the problem: I have a collection of collections, $C$, where each $c\in C$ is a collection of sets $X\subset U$. Denote $c_i$ as the i-th $X$ in $c$. Informally, I want to map all the sets in ...
6
votes
2answers
173 views

Which algorithm can I use to allocate human resources?

I have to manage shifts of a variable number of people inside several rooms for a week. Every shift must be at least 1h long and the number of hours per person for the week should be nearly the same ...
6
votes
2answers
2k views

Can we create faster sort algorithm than O(N log N)

I was thinking that we can create algorithm for sorting that will work faster than $O(N\log N)$ Let's say we have given array $A$ consisting of $N$ integers, where $N = 10^6$. Our task is to sort ...
6
votes
2answers
7k views

Algorithm to return largest subset of non-intersecting intervals

I need an efficient algorithm that takes input a collection of intervals and outputs the largest subset of non-intersecting intervals. i.e. Given a set of intervals $I = \{I_1, I_2, \ldots, I_n\}$ ...
6
votes
1answer
4k views

How to identify when to use Genetic Algorithm/Programming

I have been reading/studying on genetic algorithm/programming, and have implemented Traveling salesman problem. TSP is basically a permutation/combination problem, and I can understand how GA helps ...
6
votes
2answers
610 views

Randomized Rounding of Solutions to Linear Programs

Integer linear programming (ILP) is an incredibly powerful tool in combinatorial optimization. If we can formulate some problem as an instance of an ILP then solvers are guaranteed to find the global ...
6
votes
2answers
2k views

Shortest walk through a given subset of edges

Given an undirected weighted graph $G = (V, \{E,F\})$, how to find the shortest walk that passes through all edges $e \in E$ exactly once? I'd like to know if there is a general approach to this ...
6
votes
3answers
1k views

Interpolation Optimization Problem

I will try to give the motivation behind this problem and later the math formality. Given a grayscale image (1 Channel - M by N Matrix). Someone marks some pixels as anchors. Now, you need to ...
6
votes
2answers
1k views

A variant of the Assignment Problem

In my variant of the assignment problem I have a set $A$ of agents and a set (of possibly different cardinality) $T$ of tasks. Each agent needs to be assigned exactly $n$ or $n+1$ tasks, and each task ...
6
votes
2answers
122 views

Optimizing a join where each table has a selection

Consider the following query: ...
6
votes
1answer
406 views

How to find greatest set intersection of at least a given cardinality?

While dealing with a problem, I uncovered this subproblem: Input: A set of sets $S = \{S_1,...,S_r\}$ where $\mid$ $S_1$ $\cup$ ... $\cup$ $S_r$$\mid = n$, as well as a number $k<n$. Output: A ...
6
votes
2answers
6k views

How to find spanning tree of a graph that minimizes the maximum edge weight?

Suppose we have a graph G. How can we find a spanning tree that minimizes the maximum weight of all the edges in the tree? I am convinced that by simply finding an MST of G would suffice, but I am ...
6
votes
1answer
53 views

Perpendicular vectors out of a set

I stumbled on this problem and I wanna know if there is a better solution. There are $n$ 3d vectors with $x$, $y$, and $z$ components and I wanna find all pairs of perpendicular vectors in this set. ...
6
votes
1answer
1k views

Solving road trip problem in linear time

Consider the following problem: You are on a road trip, and there are $n$ cities along a road, labeled $1$ to $n$. Conveniently, these cities all lie on a single road, and the distance between ...
6
votes
3answers
18k views

Artificial Intelligence: Condition for BFS being optimal

It is said in the book Artificial Intelligence: A Modern Approach for finding a solution on a tree using BFS that: breadth-first search is optimal if the path cost is a nondecreasing function of ...
6
votes
1answer
81 views

Selecting vertices in a graph in an order to keep border vertices as few as possible

I am given an undirected graph. Initially all vertices are white. I need to color them black in such an order that the maximum number of vertices which are on the border between black and white ...
6
votes
1answer
169 views

Which potential function does this algorithm minimize or maximize?

Considering two sets $A, B$ containing some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^S = \min_{x' \in S \setminus \{x\}} \lVert \mathbf{x} - \mathbf{x'} \rVert$ denote the Euclidean ...
6
votes
2answers
1k views

Subset optimization problem

Consider we have a finite set $S$ with $n$ distinct elements. We want to find a subset $\{a_1, a_2, \dotsc, a_k\}\subseteq S$ ($k\ll n$) such that a function $f(a_1,a_2,\dotsc,a_k)$ is maximized. ...
6
votes
2answers
543 views

If a convex optimization problem can be NP-Hard, in what sense are convex problems easier than non-convex problems?

Being new to the OR and Optimization world, I've always assumed that a problem being convex meant that it can be solved in polynomial time. Now I am learning that a convex optimization problem can ...
6
votes
1answer
3k views

Fitting different rectangles inside a rectangle

I have a fixed rectangle of size X x Y. I also have a bunch of rectangles of different sizes. I want to check if these rectangles can fit in the larger X x Y rectangles knowing that one can rotate ...
6
votes
3answers
1k views

Why do we try to maximize Lagrangian in SVMs?

I was learning about support vector machines from MIT OpenCourseWare. I figured it out. I understand why we try to minimize $\frac{1}{2} w^2$. I just did not get why we try to maximize Lagrange ...
6
votes
1answer
321 views

How to find several rectangles with minimum area to cover the red cells

In Figure 1, (a) is the input mesh, we want to find several rectangles to cover the red cells in (a), at the same time, the sum area of these rectangles should be as small as possible. Figure 1(b) and ...
6
votes
1answer
245 views

Optimal partition of a set of pairs

Suppose we have a set $S = \{(a_1,b_1),...,(a_n,b_n)\}$ where $a_i < m$, $b_i = m-a_i$, $m \in \mathbb{Z}^{+}$, $m>2$ and $n$ is an even number greater than $3$. What is the most efficient ...
6
votes
1answer
165 views

Can all packing/covering problems be rephrased as set packing/covering problems?

Can all packing problems be rephrased as set packing problems? Can all covering problems be rephrased as set covering problems? In other words, I was wondering if set packing/covering problems are ...
6
votes
2answers
182 views

Odd cycle transversal and linear programming

Suppose we have a graph $G$ with $n$ vertices. Suppose LP is a linear programming problem where there is a variable for each vertex of $G$, each variable can take value $≥0$, for each odd cycle of $G$ ...
6
votes
1answer
147 views

Optimal partitioning of n-tuples

Motivation Recently I was trying to optimize some API calls and reduced the problem to optimization of a cumulative number of identifiers across all the requests. I put some considerable effort into ...
6
votes
1answer
972 views

Knapsack with same value

I'm wondering if there's a name/reference for the variant of knapsack problem where all items have the same value (so we only care about maximizing the number of items), but there are multiple weight ...
6
votes
1answer
374 views

Efficiently split a point cloud into two parts by a hyperplane to maximize the total sum of values associated with one part

I have the following problem in mind. Suppose we have an $n$-dimensional point cloud with $m$ points. Each point in the cloud is associated with a value $X_i,1\leq i\leq m$. I would like to use a ...
6
votes
1answer
599 views

Finding a certain prefix of a string

Let $\Sigma = \{ \sigma_1 , ..., \sigma_t \}$ and let $S$ be a string from $\Sigma^*$. Denote: $n=|S|$, that is $S$ has $n$ letters. I'd like to find the shortest prefix $T$ of $S$ such that $S$ is a ...
6
votes
1answer
86 views

Time/Space Optimal k-Subset Operator Application - Is this a named problem?

I have searched extensively and unsuccessfully for references to a combinatorial problem that arises in my work. I am hoping someone can tell me if this type of problem has a "name" and provably ...
6
votes
1answer
476 views

How to find the supremum over all the “good” (interior) polytopes for a given set of 3D points?

Let $S \subset \mathbf{R}^3$ be a set of points in 3D and let $O=(x_0,y_0,z_0)$ be the origin/point of reference. We consider a convex polytope $P$ good / interior if: $P$ is wholly contained ...
6
votes
0answers
347 views

Trying to find a human-usable method to figure out optimal round 1 openings for this game

I'm trying to figure out optimal round 1 openings for this game: http://generals.io/. For the purposes of this question, I've simplified some of the rules and mechanics of the game, and I assume that ...
6
votes
0answers
151 views

When does greediness guarantee optimality?

I was wondering if there is any theoretical results characterizing under what condition does greedy algorithm actually finds the optimal solution. Here is a motivating example. Suppose you are trying ...
6
votes
0answers
641 views

Minimal regular expression that matches a given set of words

I have a dictionary-like regular expression, an "or chain" of words, word1|word2|word3|... Unfortunately, the chain is too large. I'd like to find the minimal ...
6
votes
0answers
233 views

Is greedy minimax permutation rejecting sorting optimal?

I sketch an impractical, theoretical comparison sort. Initialize a list of all $n!$ permutations of size $n$. For each possible pair of indices $i, j$, count how many permutations would get rejected ...
6
votes
0answers
741 views

Minimum vertex-weight directed spanning tree where the weight function depends on the tree

Given a directed graph $G=(V,E)$ and a node $r\in V$, I need to grow a tree $T$ rooted at $r$ that has a minimum weight and spans all reachable nodes in $G$. The weight function assigns a non-...
5
votes
4answers
1k views

Does Thompson's algorithm produce optimal NFAs?

I'm using Thompson's algorithm to convert from a regular expression to a NFA. Is Thompson's algorithm guaranteed to always output a minimal NFA, i.e., a NFA with the smallest possible number of ...
5
votes
2answers
7k views

The stable marriage algorithm with asymmetric arrays

I have a question about the stable marriage algorithm, for what I know it can only be used when I have arrays with the same number of elements for building the preference and the ranking matrices. ...
5
votes
4answers
618 views

Are there any optimization problems in P whose decision version is hard?

Normally to show that an optimization problem is hard, we show the corresponding decision version of the problem is hard. However, is this sufficient to support the conclusion? Does there exist any ...
5
votes
2answers
132 views

Unknown notation “$e^T$” in a machine learning paper

I'm trying to understand the material in "A Dual Coordinate Descent Method for Large-scale Linear SVM" by Hsieh et. al. (link to paper) There is an equation for the Dual form of an unconstrained ...
5
votes
3answers
2k views

Portfolio rebalancing algorithm

Quite possibly this problem is already solved and has it's own name, but I was unable to find any directions. So, I have a portfolio with some items in it: | Name | Price | Amount | Share | |------...

1 2
3
4 5
29