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Questions tagged [matroids]

16 questions
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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 algorithms to find optimal solutions. The…
Nicholas Mancuso
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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 have roughly understood what matroids are. But how…
Imposter
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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 to find $$\min_{x \in \mathbb{Z}^2} f(x)$$ where…
ChubbyRuby
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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 an element with maximum weight among all…
Erel Segal-Halevi
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Trying to figure if greedy algorithm is a matroid (or greedoid)

I know that if I can express a problem as a (weighted) matroid M = (E, S) or a greedoid then I can assure that there is an algorithm which will give me the optimal solution. For example for matroids it goes something like: T = {} sort(E)…
aram
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A task-scheduling problem as a matroid (CLRS book)

I don't understand very well section 16.5 of the 3rd edition of the famous Introduction to Algorithms book, known as CLRS. It defines the problem of scheduling unit-time tasks with deadlines and penalties for a single processor with the following…
vvaltchev
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Is the intersection of $k \geq 3$ graphic matroids in P?

It is known that intersection of three general matroids is NP-hard (wiki), which is done via reduction from Hamiltonian cycle. The reduction uses one graphic matroid and two connectivity matroids. A special case of a problem I am working on can be…
Matej Konecny
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Difficulty in understanding the proof of the lemma : "Matroids exhibit the optimal-substructure property"

I was going through the text "Introduction to Algorithms" by Cormen et. al. where I came across a lemma in which I could not understand a vital step in the proof. Before going into the lemma I give a brief description of the possible prerequisites…
Abhishek Ghosh
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Activity Selection and Matroid Theory

Many people on different articles suggests that if an optimization problem has a greedy solution, the underlying structure must have matroid property. I was trying to understand this. So far, I was able to prove that for, Maximum sum of m…
Shakil Ahamed
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Can LP for matroid polytopes be solved using the greedy algorithm?

For general linear programming (LP), i.e. optimization of a linear objective over a general polyhedron, to the best of my knowledge/recollection one can use the simplex algorithm (or hypothetically, although it's not done often in practice, the…
Chill2Macht
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What is a good resource to learn about oriented matroids in the context of digraphs and optimization?

I am interested in oriented matroids in the context of directed graphs and optimization. Unfortunately, I know very little of the topic. Is there a book, article or a resource that serves as a good introduction to oriented matroids, especially in…
Juho
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A $log(k)$ algorithm for the matroid secretary problem

I'm reading the following article that presents a $log(k)$ algorithm for your secretary problem. I'm in the analysis section at the left part of page 5 there is the following claim: $B^*$ is a set with elements $x_1,...,x_k$ with respective…
Belgi
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Rank of a graph in matroid theory

I was going through the concept of graphs as matroids and I came upon the rank of a graph. Wikipedia lists it as $n - c$, $n = |V|$, $c =$ # of connected components. I do understand rank and nullity of matrices, and graphs when expressed in their…
Ash Ketchum
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The Greedy Algorihm for Matroids works for maximisation and minimisation

I am working on the following exercise: Let $(S,\mathcal{F})$ be a matroid and let $c:S \rightarrow \mathbb{R}$ be a weight function on $S$. Find an algorithm that solves the following problem: MAXIMISATION PROBLEM FOR INDEPENDENCE SYSTEMS (MAX…
3nondatur
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Does the existence of a matroid structure imply that the greedy algorithm is optimal?

I was going through the topic of matroid structures for the problems like Activity selection ,minimum spanning tree. I also came to know how to solve if a problem exhibits matroid structure. The statement says that problems that have matroid…
Crypto_Research
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