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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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### Pick a subgraph that maximizes the total cost of min-spanning tree among all subgraphs of the same size

There is a complete graph $G$ with $n$ vertices and each edge has a distinct weight. Is there an efficient (not necessarily optimal) algorithm to select $k$ vertices from the graph $G$, such that the ...
165 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 ...
67 views

### Finding the number of ways to partition $\{1,…,N\}$ into $P_1$ and $P_2$ such that $sum(P_1) = sum(P_2)$ for a given $N$

I am trying to think of how to optimize the following problem: Let $S = \{1,2,...,N\}$. How many ways can $S$ be partitioned into non-empty subsets $P_1$ and $P_2$ such that $sum(P_1) = sum(P_2)$? I ...
336 views

### Definition of $\alpha$-approximation

I know this question is trivial. But I am looking for a concise formal definition of $\alpha$-approximation. Is it correct to say that "An algorithm is an $\alpha$-approximation to problem $X$, if ...
178 views

### Distributing resources for maximum gain

This feels like it would be a well researched (or solved) problem, but I can't find the right words to search for it. Suppose there is a collection of shared resources, and a collection of possible ...
128 views

### Discrete assignment problem with penalties

I came across a problem were you have to plan an optimal assignment pattern. Let's say you have $j=1,\ldots,n$ tasks during $i=1,\ldots,m$ time periods. It's an single agent problem where we have to ...
778 views

### Closed form solution for optimization problem

Consider the problem of finding the real-valued matrix $C$ so that $$\|S-AC\|_F^2\qquad(1)$$ is minimal. ($S$ and $A$ are real valued matrices and $_F$ denotes the Frobenius norm). This problem has ...
224 views

### Greedy Algorithms for Non-monotone Submodular Maximization with Cardinality Constraints

Does any approximation algorithm exist for maximization non-monotone submodular functions that might have negative values or be unbounded below? Fact 1: For monotone submodular functions, Nemhauser, ...
2k views

### Using dynamic programming to maximize work done

Say that there are $n$ days and there is $x_1, x_2, ...,x_n$ amount of data to process on each day. Your computer can process $s_1$ amount of work on the first day since rebooting your computer, $s_2$ ...
816 views

### Minimizing inversions in an array with a single swap

This was asked in the (very) recently concluded Hackerrank Worldcup. Paraphrased: Given a permutation $a$ of integers from $1$ to $N$, how can I minimize the number of inversions by a single swap ...
575 views

### Using a computer algebra system to optimize mathematical expressions

This is something I've been wondering for years. Software like Mathematica is great at manipulating expressions into simplified, factorized, and other forms. I'm wondering if there's a way, ...
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### Algorithm(s) for creating balanced 5-a-side teams?

I play 5-a-side soccer twice a week. 10 people play each game out of a pool of about 30 people who are giving the opportunity to play as it's first come first serve. Some people play more often than ...
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 ...
113 views

### What is the global function we are trying to Optimise with Clustering Algorithms?

I am doing some reading (and implementation) of some Clustering Algorithms. First I started with the well known K-Mean algorithm and implemented it directly from a paper. Got a kind of decent ...
173 views

### Genetic algorithm: What is the expected number of strings that are explored?

My question concerns genetic algorithm searching along bit strings. Given: $N$ = population size $l$ = length of bit strings $p_c$ = probability that a single crossover occur (double crossover never ...
183 views

### Formulating a linear program s.t. only extreme point solutions are found

If there are many solutions to a linear program s.t. the objective function is minimized/maximized (= optimal solutions are on an edge of the polytope), how can I force an LP solver to find only an ...
37 views

### Why does it take so long to prove optimality when warm-starting from optimal solution

So I'm solving bigger instances of some binary-linear-program using cplex. The formulations of the problem I am using is integer friendly, meaning nearly all of my instances can be solved at the root ...
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### Find ellipsoid that contains intersection of an ellipsoid and a hyperplane

I have an $n$-dimensional ellipsoid $E$ and a hyperplane $H$. This hyperplane cuts $E$ into two parts: $E_1$ and $E_2$ (whose disjoint union is $E$). I want to find another ellipsoid $E'$ that has ...
306 views

### Finding a maximum-weight base of a a matroid, in reverse

Given a weighted matroid with positive weights, we can find a independent set with a maximum weight with a greedy algorithm: Start with an empty set (by definition of matroid, it is independent). Add ...
363 views

### Linear optimization or Constraint Satisfaction Problem with food

I was hoping someone could point me in the right direction in terms of what type of problem I am describing here so I can research it. My initial thought is that it is some form of Constraint ...
554 views

### Searching the best trading route - algorithm

Imagine a village with people trading goods. Each person has his own offer in this format: Giving amount a of good b for amount <...