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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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8 votes
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How to find the shortest path from some vertex in set $S$ to set $S'$

If i have a graph $G=(V,E)$, a subset of vertices $S \subset V$ and a second set of vertices $S' \subset (V\setminus S)$, what is the best way to find the shortest path connecting the two sets? That ...
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7 votes
1 answer
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 ...
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  • 171
6 votes
1 answer
780 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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5 votes
2 answers
8k views

What TSP variant doesn't return to start point?

For my case I have starting point and several cities. I want the shortest route to visit all cities without returning starting point. I have read several TSP algorithm and all include the return a ...
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4 votes
2 answers
530 views

Dynamic Shortest Path with Linear Programming

Consider a grid with $x=5$ columns, $y=5$ rows, and $T$ timesteps. There are $N=2$ agents in this grid, which can move vertically or horizontally. The positions of each agent $x$ at timestep $t$ is ...
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11 votes
3 answers
2k views

Algorithms for minimizing Moore automata

Brzozowski's algorithm can be extended to Moore automata but its time complexity is exponential in general. Is there any other algorithm for minimization of Moore automata? What are the running times ...
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10 votes
1 answer
775 views

Can we find k shortest paths between all pairs faster than solving the pairwise problem repeatedly?

I want to produce $k$ shortest path ($k$ would be less than 10) between all pairs in a graph. The graph is (actually a subway map): positively weighted undirected sparse with about 100 nodes My ...
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8 votes
1 answer
1k views

Linear time algorithm for finding $k$ shortest paths from $s$ to $t$

Definition. Given a graph $G=(V,E)$ and two vertices $s$ and $t$, the $k$-shortest-paths problem is finding the $k$ shortest simple paths between $s$ and $t$ in $G$. Note that the length of ...
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6 votes
1 answer
200 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 ...
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  • 221
5 votes
1 answer
104 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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3 votes
1 answer
527 views

Is this problem NP-hard?

Good day. Subset sum selection problem is NP-hard. I trying to solve following problem: Input: a grid NxN and subset size K and radius R. Every entry in grid contains a value. Solution: subset of ...
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  • 259
2 votes
1 answer
4k views

Assignment based on ranked preference

Assume that there are n students, who have to be evenly assigned to m groups. For every student, a preference ranking of of the <...
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1 vote
2 answers
2k views

Genetic Algorithm Minimum Population Size

Is there a minimum limit to a pool (population) size when using the genetic algorithm to solve an optimization problem? For example a population of size 2.
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1 vote
2 answers
207 views

Literature on network-flow (optimization) approximation algorithms

I've been searching on literature on approximation algorithms in the context of network-flow problems (optimization) to finish my bachelor degree. However, I have been looking in several well-known ...
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  • 491
7 votes
1 answer
65 views

Is there a more up-to-date / wider-scope version of the 'Compendium of NP Optimization Problems'

When I was studying Comp Sci, we had Garey & Johnson as a course textbook, with a large collection of NP-Complete problems. But by that time you could also have a look at the Compendium of NP ...
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  • 937
7 votes
1 answer
207 views

Sorting an unordered pile of items into drawers with minimal drawer movements

A while ago, I was doing my laundry late at night. When I brought my laundry back to my dorm, I started to put it away. My wardrobe is set up as follows: My drawers are categorized by the type of ...
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  • 325
7 votes
1 answer
1k 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 ...
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5 votes
2 answers
803 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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5 votes
1 answer
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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  • 51
5 votes
1 answer
265 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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  • 1,320
5 votes
2 answers
115 views

Detecting conservation, loss, or gain in a crafting game with items and recipes

Suppose we're designing a game like Minecraft where we have lots of items $i_1,i_2,...,i_n\in I$ and a bunch of recipes $r_1,r_2,...,r_m\in R$. Recipes are functions $r:(I\times\mathbb{N})^n\...
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  • 153
5 votes
1 answer
3k views

Is the set partitioning problem NP-complete?

I know that the set partitioning problem defined like this: Given $$S = \left\{ x_1, \ldots x_n \right\}$$ find $S_1$ and $S_2$ such that $S_1 \cap S_2 = \emptyset$, $S_1 \cup S_2 = S$ and $\...
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5 votes
3 answers
1k 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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4 votes
1 answer
544 views

Maximize ratio of sums

I have a $2 \times n$ matrix of positive integers, where the elements are denoted by $a_{ij}$ for all $i$ in the set $\{1,2\}$ and for all $j$ in the set $\{1,\ldots,n\}$. I would like to select a ...
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  • 970
4 votes
1 answer
317 views

Fixed size set to contain the maximum number of given sets

I asked this question in SO here I have about 1000 sets of size <=5 containing numbers 1 to 100. {1}, {4}, {1,3}, {3,5,6}, {4,5,6,7}, {5,25,42,67,100} ... ...
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  • 173
4 votes
2 answers
181 views

Largest weight-limited connected subgraph: NP-complete?

When playing Terra Mystica, it might be useful to predict how many spades you will get throughout the game, and use this information to decide where to build, such that you stand a good chance of ...
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3 votes
2 answers
749 views

Why are decision problems easier than the equivallent optimization problems?

Suppose that we have an optimization problem defined as follows: $OPT$ = Given an input string defining a set of feasible solutions $F$ and an objective function $f$, find $x\in F$ maximizing $f(x)$ ...
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  • 599
3 votes
2 answers
891 views

Which is the proper algorithm to figure repeated substrings of longest total length?

Given a string $S$ and a substring $T$, define the value of $T$ to be the total length (in characters) of all non-overlapping instances of $T$ in $S$. In other words, the value of $T$ is the length ...
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  • 131
3 votes
2 answers
398 views

Minimising sum of consecutive points distances Manhattan metric

I have two sets $X$ and $Y$ of 2-dimensional points. The points are floating point numbers. The objective is to sort them in such way that sum of differences in distances of consecutive sorted points ...
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  • 9,325
3 votes
2 answers
427 views

Solving/Optimizing a linear system in a finite field (Z/2Z)

I'm trying to solve the following optimization problem. A is a rectangular matrix with coefficients in the finite field Z/2Z (size less than 1000 X 1000). I have a system of the form A.X = Y (X and Y ...
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  • 335
3 votes
0 answers
1k views

Finding the n-best items in a 0/1 Knapsack

I'm trying to understand why an alternate formula for finding the best $p$ items in a 0/1 knapsack with $n$ items isn't working. The formula was proposed by @Carlos Linares López in this answer: ...
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2 votes
1 answer
408 views

efficient algorithm for min cut with specified number of vertices

Consider a graph with vertices $V$ and edges $E$. The standard version of the min cut problem is to find the partition of $V$ into a (non-empty) subset $C$ and its complement $\bar{C}$ so as to ...
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  • 123
2 votes
2 answers
1k views

How to use dynamic programming to solve this?

Here is the question: suppose we are given x cents, the amount we want to pay, and a 6-tuple (p, n, d, q, l, t) that represents respectively the number of pennies, nickels, dimes, quarters, loonies ...
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  • 1,091
2 votes
1 answer
1k views

Is it possible to make data structure that will find MEX and support modification queries

Let's say we have given array of $n$ elements, now we want to create data structure that will allow us to get the MEX of its elements, MEX (Minimum EXcluded) meaning the smallest positive integer that ...
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  • 1,398
2 votes
1 answer
4k views

Shortest path from that passes through a set of edges once

Given a graph with weighted edges. How to find the shortest path from vertex $A$ to vertex $B$ that passes through a set of edges $X$ at most once? $X$ can be big. Slow solution: Finding shortest ...
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  • 123
1 vote
1 answer
3k views

Board cutting problem

To cut a wooden board, a sawmill charges proportional to the length of the board. The cost of cutting a single board into many smaller boards will thus depend on the order of the cuts. As an example, ...
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1 vote
1 answer
169 views

What makes an MILP problem solvable?

Knapsack problems, Assignment problems can all be expressed as (MILP) mixed integer linear programs. MILP is NP-complete. But Knapsack problem is solvable in pseudo-polynomial time using dynamic ...
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  • 153
1 vote
0 answers
615 views

Streaming Knapsack Problem

I want to implement efficiently "streaming Knapsack" problem in java. The problem is I have a stream input of integer data coming continuously for example -1, 2, 9, 5, 5, 11, 1 -3,... The question ...
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  • 2,238
0 votes
2 answers
2k views

Interval Scheduling Problem with Three Resources

So I am trying to figure out what kind of algorithm I would use if I wanted to implement the ISP with a predefined number of resources. Consider this example interval scheduling problem. Say there ...
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9 votes
3 answers
1k views

How to solve an arrangement problem at the Archive Nationale of France using graph theory?

Good evening! I'm actually doing an internship at the Archives Nationales of France and I encountered a situation I wanted to solve using graphs... I. The dusty situation We want to optimize the ...
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8 votes
2 answers
8k views

Algorithm to find optimal currency denominations

Mark lives in a tiny country populated by people who tend to over-think things. One day, the king of the country decides to redesign the country's currency to make giving change more efficient. The ...
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  • 12.6k
7 votes
1 answer
473 views

Given 2 sets of n points: minimize sum of traveled distances

I am given two sets $S, T$ each of $n$ points in $\mathbb{R}^k$, I want to find a bijection $a : S \rightarrow T$, such that $$\sum_{s \in S} d(s, a(s))$$ gets minimized, with $d$ being the Euclidean ...
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6 votes
1 answer
2k views

Which algorithm can calculate an optimal allocation of students to projects?

I am trying to find an algorithm which calculates an optimal and stable allocation of $n$ students to $m$ projects, where each student strictly ranks all projects by preference. The available projects ...
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  • 63
5 votes
4 answers
2k 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 ...
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  • 219
5 votes
1 answer
718 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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  • 153
5 votes
2 answers
182 views

Constraint violation and efficiency in search

It seems that (in a broad sense) two approaches can be utilized to produce an algorithm for solving various optimization problems: Start with a feasible solution and expand search until constraints ...
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4 votes
0 answers
71 views

Find the largest interior rectangle composed of partitioning rectangles

Suppose a rectangle $R$ is partitioned into more than one smaller rectangles of positive area. What is the fastest algorithm to find the largest rectangle strictly within the all-enclosing rectangle $...
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  • 187
4 votes
2 answers
3k views

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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  • 431
4 votes
2 answers
2k views

Maximum sum subset of an array with an extra condition

We are given numbers $n \leq 200$, $k \leq 10$ and an array of $3n$ positive integers not greater than $10^6$. Find the maximum possible sum of a subset of elements of this array, such that in every ...
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  • 187
4 votes
1 answer
187 views

What is the significance of the vector dimension in semidefinite programming relaxations?

Let's say that we want to design a semi-definite programming approximation for an optimization problem such as MAX-CUT or MAX-SAT or what have you. So, we first write down an integer quadratic ...
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  • 329