# Questions tagged [greedy-algorithms]

Questions about algorithms that make at each step the locally optimal choice.

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26k views

### How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
27k views

### When can a greedy algorithm solve the coin change problem?

Given a set of coins with different denominations $c1, ... , cn$ and a value v you want to find the least number of coins needed to represent the value v. E.g. for the coinset 1,5,10,20 this gives 2 ...
1k views

### How fundamental are matroids and greedoids in algorithm design?

Initially, matroids were introduced to generalize the notions of linear independence of a collection of subsets $E$ over some ground set $I$. Certain problems that contain this structure permit greedy ...
1k views

### How to use a greedy algorithm to find the non-decreasing sequence closest to the given one?

You are given n integers $a_1, \ldots, a_n$ all between $0$ and $l$. Under each integer $a_i$ you should write an integer $b_i$ between $0$ and $l$ with the requirement that the $b_i$'s form a non-...
11k views

### Correctness-Proof of a greedy-algorithm for minimum vertex cover of a tree

There is a greedy algorithm for finding minimum vertex cover of a tree which uses DFS traversal. For each leaf of the tree, select its parent (i.e. its parent is in minimum vertex cover). For each ...
187 views

### Fixed-length decision-tree-like feature selection to minimize average search performance

I have a complex query $Q$ used to search a dataset $S$ to find $H_\text{exact} = \{s \in S \mid \text{where$Q(s)$is True}\}$. Each query takes on average time $t$ so the overall time in the linear ...
8k views

### Find non-overlapping scheduled jobs with maximum cost

Given a set of n jobs with [start time, end time, cost] find a subset so that no 2 jobs overlap and the cost is maximum. Now I'm not sure if a greedy algorithm will do the trick. That is, sort by ...
878 views

### Greedy choice and matroids (greedoids)

As I was going through the material about the greedy approach, I came to know that a knowledge on matroids (greedoids) will help me approaching the problem properly. After reading about matroids I ...
513 views

### Balanced weighting of edges in cactus graph

Given a cactus, we want to weight its edges in such a way that For each vertex, the sum of the weights of edges incident to the vertex is no more than 1. The sum of all edge weights is maximized. ...
3k views

### Knapsack Greedy Approximation: Worst Case

I am currently studying approximation algorithms and I have run into an issue with a study problem. The approximation algorithm is for the general Knapsack problem, and it proposes a greedy approach, ...
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 ...
287 views

### The heaviest induced subgraph problem

I am interested in such a combinatorial problem: given a graph $G=(V, E)$ and a weight functions $w_v: V \mapsto R$, and $w_e: E \mapsto R$ we are asking about such a induced subgraph $G' = (V', E')$ ...
1k views

### Please explain a greedy algorithm in a naive manner [closed]

I am a beginner in the topic of algorithms. I have a query about Greedy Algorithms. From what I understand, if there is a function and we are supposed to find its maxima/minima, if we find the local ...
270 views

### Is there a polynomial time algorithm to determine whether an 'up down' language is 'emptible'?

Definitions: An up down language is a language whose alphabet is a set of pairs, but not characters, of two characters, where the one character in the pair is the opposite of the other character in ...
144 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 ...
6k views

### Greedy and backtracking solutions to an arrangement problem with constraints

I'm revising for my finals. I have found a pattern in past papers in terms of a recurring question, reworded coming up every year. But I've no idea what the marker actually wants... I've asked class ...
948 views

### Counterexample to this modified Dijkstra's

In class, we were given the following problem: We are given a directed graph G = (V, E) on which each edge (u, v) ∈ E has an associated value r(u, v) which is a real number in the range 0 ≤ r(u, ...
519 views

### Do all greedy algorithm produce just the first solution, no matter how bad it is?

In all the exampls of the greedy algorithms I've seen so far, such as activity selection problem and unit-sized set coverage problem, the algorithm is usually very simple and intuitive and returns the ...
235 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, ...
449 views

### Variations of Greedy Algorithm

What is the definition of an "orthogonal greedy algorithm"? What is the definition of a "relaxed greedy algorithms"? Can you give an example to illustrate how these notions differ from ordinary ...
16k views

### Time complexity of a backtrack algorithm

I've developed the following backtrack algorithm, and I'm trying to find out it time complexity. A set of $K$ integers defines a set of modular distances between all pairs of them. In this algorithm, ...
1k views

### Greedy algorithms: Minimum sum number pairing

Given $n$ real numbers (where $n$ is even) find a pairing which minimizes the maximum sum of a pair. I think the optimal pairing is obtained by sorting the original set, pairing the first element ...
1k views

### Greedy proof: Correctness versus optimality

I am really confused after surveying a bunch of material online about correctness versus optimality proof for greedy algorithms. Some website even uses both correctness and optimal in the same ...
296 views

### Minimizing Cost by minimizing delay

There is a complete binary tree with its leaves as components of some system The values from one node to another gives propagation time for a signal to propagate from one junction to another For the ...
430 views

### Why do the swap step in Prim's algorithm for minimum spanning trees?

I was watching the video lecture from MIT on Prim's algorithm for minimum spanning trees. Why do we need to do the swap step for proving the theorem that if we choose a set of vertices in minimum ...
331 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 ...
421 views

### Greedy algorithm correctness proof for “Elegant Permuted Sum” (UVa 11158)

Given a sequence of $2 \leq n \leq 50$ numbers $s = (s_1,s_2,...,s_n)$, find a permutation $a = (a_1,a_2,...,a_n)$ of $s$ such that $$\sum_{i=1}^{n-1} |a_i - a_{i+1}|$$ is maximized. I found many ...
485 views

### Issues with using greedy algorithm (Interval scheduling variant)

I am trying to solve a problem of finding incompatible jobs set using greedy algorithm. However, I am not sure if greedy algorithm can solve this problem or I need to perform another approach. I have ...
230 views

### An algorithm for a minimization problem, How to minimize the wasted length of combination of multiple items with different length and number

Suppose there is an unlimited number of pipes, each has length $x$ meters. There is a list of requirements of pipes with shorter length than $x$. The number of these items are also given. For example ...
445 views

### 2 approximation algorithm for the single machine scheduling problem

We are given one machine and $n$ jobs that we want to process. For the $n$ jobs we have the following data: $r_{1}, ... , r_{n}$ are the release times $p_{1}, ... , p_{n}$ are the processing times ...
93 views

### How to cluster similar objects into fixed size groups?

I have $n$ people each of which can meet on certain days of the week. I want to group them into $\frac{n}{k}$ groups of size $k$ such that all people in a group can meet on a day. eg - Suppose there ...
204 views

### A general algorithm for greedy algorithms

I have been refreshing on greedy algorithms as an algorithm design technique. I have read many sources for an explanation of what a greedy algorithm is, because I would like to put together a general ...
802 views

### set with maximum sum consisting of mutually co-prime numbers

Definitions. Let $n$ be a natural number and $S$ be a subset of distinct natural numbers all less than $n$, and mutually co-prime. Then find the maximum sum the set $S$ can have. Example. Let $n=10$, ...
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### Prove that the greedy algorithm to remove k digits from a n-digit positive integer is optimal

Given a positive n-digit integer, such as 1214532 (n=7), remove k digits (for example k=4) such that the resulting integer is the smallest one. A greedy algorithm for this would keep removing digits ...
63 views

### Maximize number of museums visited in a day

Given a list of museums, their opening hours and time needed to visit each, make a schedule such that a tourist visits maximal number of museums in a given day. Suppose that no time is needed in ...