Questions tagged [optimization]

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

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Why does Kadane's algorithm solve the maximum sub-array problem?

I've tried to solve a exercise 4.1-5 in algorithm book "Introduction to algorithms". it is about Maximum sub-array problem, which is an algorithm that determines the greatest sum of sub-array A[i], ...
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1answer
234 views

Moving an edge in a weighted tree to maximize longest path length

Let $G$ be a undirected edge-weighted tree, where all edge weights are positive. A move of an edge $\{u,v\} \in E(G)$ is the operation of deletion of $\{u,v\}$ and the addition of a new edge $\{x,y\}$,...
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150 views

Optimization over convex combinations in a circle

Consider the following situation: given a triangle $ABC$ inscribed in a circle, define $f$ as the product $$f(P) = d(P, A) \; d(P,B) \; d(P,C)$$ where $P$ is a point on the circle and $d$ are ...
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80 views

Finding minimal and complete test sets for circuits

I have been playing around with analysis of circuits and am trying to generate test vectors. In order to exercise the circuit in the manner I require, I need a vector that includes every change in the ...
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1answer
215 views

Algorithms for logical synthesis of multiple output bits?

Karnaugh maps and the Quine–McCluskey algorithm can be good choices for coming up with fairly minimal logical expressions that match the requirements of a truth table. What if I have a situation ...
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1answer
153 views

Fast solution for a combinatorial maximizaton problem

You are given a natural number n (n<20). We construct the set S from all binary numbers with n bits. We call two numbers "compatible" if they don't have any common substring of length n-1 (...
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1answer
594 views

Finding set of disjoint sets with additional value optimization

I've got a set $Q$ of pairs $[S, v]$ where $S$ is a nonempty set and $v$ is a value ($v \in \mathbb{N}_{+}$). I need to find a subset $R$ of $Q$ with following properties: Sum of all $v$'s is maximum ...
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1answer
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Seating Chart Optimization

I'm trying to find an algorithm to solve the seating chart problem. The goal is to place pepole at one (or multiple) tables such that the overall happiness is maximized. Each seat has neighbors. A ...
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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. ...
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1answer
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Optimization problem vs decision problem - reduction

Assume we have an optimization problem with function $f$ to maximize. Then, the corresponding decision problem 'Does there exist a solution with $f\ge k$ for a given $k$?' can easily be reduced to ...
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186 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 ...
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110 views

Find a strategy to evade hungry lions on the real line for the longest time

This is an interview question I was asked, which I don't know how to approach. I would appreciate pointers to algorithms I should look up. You are placed on the real line, and there also are $K$ ...
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Finding k-nearest neighbors to a set of nodes in a large graph

Given a large graph $G=(V,E)$, a set of nodes $S\subseteq V$, the problem is finding the $k$-nearest nodes in $V$ to the nodes in $S$. Given a pair of nodes $(u,v)$, the distance $d(u,v)$ between $u$ ...
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1answer
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How can I restructure matrices to have non-zero elements close to the diagonal?

I have a matrix $C \in \mathbb{N}^{n \times n}$. Semantically, it is a confusion matrix where the element $c_{ij}$ denotes how often members of class $i$ are predicted by a given classifier as members ...
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38 views

Find expression with minimal distance to target

I will start of with an informal example and give a more formal problem definition later. Say I have a finite set of positive real values: $\{2.3, \pi, 4.382, 0.3\}$. Using normal addition and ...
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324 views

Transition coverage for a DFA

Let $G$ be a directed graph, with a single source node $s$. I want to find a collection of paths that cover every edge of $G$ (i.e., every edge of $G$ appears in at least one of these paths), where ...
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826 views

Decision vs Optimization version for Problems of two Parameters

Let's say I have an optimization problem called $k$-foo which asks for a solution of size $k$ minimizing some quality criterion. Now the corresponding decision problem $foo(M)$ would be: Is there a ...
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3answers
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Min cost max flow in bipartite run time

I have a bipartite graph with $|E|=O(|V|^2)$, a super-source and a super-sink. I am looking for the min-cost max-flow (the max-flow of all possible max-flows that has the minimum cost). For the sake ...
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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 ...
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247 views

Maximize product of sum of two subset

Given two sets $A = \{a_1, a_2, \dots, a_n\}$ and $B = \{b_1, b_2, \dots, b_n\}$, both consist of positive numbers, this problem is to find a subset $S$ in $\{1, 2, \dots, n\}$ to maximize $$ \left(\...
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94 views

Location Selection Algorithm in Solar Engineering

This is a practical problem in energy generation with heliostats. We have a number of heliostats basically forming the shape of a doughnut. The facility needs to deploy hubs on those heliostats. One ...
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191 views

What data structure might this game use?

This question is not about game development or about actual implementation details. I was playing Little Alchemy yesterday. (Warning: Productivity hazard.) You start with the four classical ...
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1answer
86 views

Coercing a list of nodes into the most probable tree

Suppose that we have an RTF document which contains sections and sub-sections. The sections and subsections all have headings that are visually marked up (e.g., bold and italic), but the document ...
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81 views

Complexity of covering subset of the monoid $(\{0,1\}^n, \text{OR})$

(At the very bottom of this, I will shortly describe the motivation for this question.) Assume we have a commutative monoid $(G,\circ)$, i.e. a set $G$ with a commutative binary operation $\circ$ ...
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79 views

Global Optimization of a well-defined function with gradient information

I try to minimize the function $$ f(x_1, …x_n)=\sum\limits_{i}^n-a_i\cos(4(x_i-b_i)) +\sum\limits_{ij}^\text{edge}- \cos(4(x_i-x_j)), \quad x_i,b_i\in (-\pi, \pi)$$ where $\sum\limits_{ij}^{edge}$ ...
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300 views

How can I efficiently find the optimal order to apply special offers to a shopping cart?

Given a list of items which represent items in a shopping cart, and a list of available special offers which replace one or more regular items to lower the cost of those items, how can I decide the ...
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2k views

Variation of Set Cover Problem: Finding a maximum-sized collection of disjoint set-covers

I have the following problem, which seems to be similar to Set Cover. We are given a set $U$ of elements (the universe, e.g., $U=\{1,2,3,4,5\}$). We're also given a set $S$ of subsets (e.g., $S=\{\{1\...
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654 views

Dividing a weighted planar graph into $k$ subgraphs with balanced weight

I've been looking for an algorithm which divides an undirected, weighted, planar and simple graph into $k$ disjoint subgraphs. Here, the graph is sparse, $k$ is fixed, and there are no negative edge ...
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89 views

Specific quadratic 0-1 knapsack problem solvable in linear time?

I am interested in a simple variant of the quadratic knapsack problem. Let $\{w_1, \ldots, w_n\} \in \{0,1\}$ be $n$ weights and $\{v_1, \ldots, v_n\} \in \mathbb{R}$ be $n$ values. Furthermore, ...
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Can all types of problems be converted to decision problems?

We know all optimisation problems can be converted to decision problems. Is that true for search problems, counting problems and function problems as well? Description of the types of problems is ...
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132 views

Approximate Weighted Partial Max SAT

Given a Weighted Partial Max SAT problem (WPM-SAT) - are there generally used algorithms or techniques to generate 'approximate' solutions, which are not necessarily optimal, but found faster than ...
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417 views

How to apply ant colony optimization to the TSP but repeating nodes and edges

I'm learning the Ant Colony Optimization Algorithm and I would like to apply it to a variation of the TSP problem (find the path that start from a node, crosses all nodes and finish in the initial ...
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41 views

Cuttings sticks in congruent equal sharing

You have $n$ congruent sticks (they have the same length). You want to divide them equaly among $m$ friends. To avoid envy, each friend should receive congruent parts, that is, the set of cutted ...
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678 views

What is the optimal way to cut B chocolate bars to share equally between N people?

What is the optimal way to cut B chocolate bars to share equally between N people? Here is an example of different cuts for B = 5 chocolate bars and N = 6 people. Strategy 1: cut each chocolate bar ...
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Linear functions of matrix exponential

Given a matrix $A$ and a vector $v$, I'm aware there are efficient algorithms for computing $e^Av$, where efficient means significantly faster than computing $e^A$ and multiplying by $v$. For a ...
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43 views

Authors of Complementary Slackness

Who were the first researchers to prove the Complementary Slackness condition for linear programming? I believe that strong optimality was proved by Gale, Kuhn, and Tucker in 1951, but I couldn't ...
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Rate Pooling Optimization Algorythim

I have thousands of wireless LTE hotspots. Each month I need to assign each hotspot a rate plan. Each hotspot uses some amount of data in a month (represented in megabytes). Each rate plan has some ...
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142 views

A matrix rank problem over finite fields

I have already asked a similar question here, but since I have not got an acceptable answer, I decided to ask a simpler version of the question here. Let $M|\mathbf w$, where $M$ is a matrix and $\...
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39 views

Does it make sense to examine the dual of a feasbility problem?

Consider a standard feasibility problem. The goal is to examine the state of feasible solutions for $Ax=b$ to find an $x$ that satisfies some property. Does the dual of this problem tell us anything ...
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1answer
517 views

Shortest path in a known room for a Roomba

I had an interview question once which asked for an algorithm to ensure a Roomba vacuum cleaner visited/vacuumed every "cell" in an unknown shape/size room with unknown obstacles. Depth first search ...
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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. ...
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4answers
552 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 ...
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2answers
568 views

Suboptimal Solution for a combinatorial problem

I have a cost function $f(X)=\|\hat{X}-X\|_2$ to minimize which depends on a $s\times s$ matrix $X$ where $\hat{X}$ is given and $\|X\|_2=\big(\sum_{i,j}x_{ij}^2\big)^{1/2} $. This matrix $X$ is ...
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Does local optimization in a genetic algorithm decrease diversity?

I want to create a hybrid genetic algorithm for a project to solve really high dimensional problems (1000+) One of my ideas is to incorporate a local optimization method within GA so each individual ...
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367 views

Linear Path Optimization with Two Dependent Variables

Alright, so this is a fairly interesting problem I have but also slightly difficult to explain so I will try my best. There are two runners on a line that goes from $x=0$ to $x=100$. The two runners ...
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Knight on a chessboard

This is a Hackerrank challenge $Knight$ is a chess piece that moves in an L shape. We define the possible moves of $Knight(a,b)$ as any movement from some position $(x_1, y_1)$ to $(x_2, y_2)$ ...
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1answer
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Find maximum distance between elements given constraints on some

I have a list of numbered elements 1 to N that fit into positions on a number line starting with 1. I also have constraints for these elements: The element 1 is in position 1, and element N must be ...
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3answers
947 views

Assign m agents to N points by minimizing the total distance

Suppose we have $N$ fixed points (set $S$ with $|S|=N$) on the plane and $m$ agents with fixed, known initial positions ($m<N$) outside $S$. We should transfer the agents so that in our final ...
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1answer
883 views

Minimize number of circles to cover set of points

I'd like to minimize the number of circles $C$ to cover a set of points $P$ in 2D. There is no fixed radius however each circle must contain exactly $N$ points. The circles may overlap. A point may ...
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1answer
78 views

Unfeasible linear program becomes feasible if a variable is removed

Apologies, not a computer scientist by trade but I'm playing with linear programming these days. Let $\{x_i\}$ be $N$ optimization variables with bounds $$l_i \leq x_i \leq u_i$$ I'm interested in ...