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I'm reading Schrijver's text on combinatorial optimization [1] in order to learn more about matroids. In his definition of a binary matroid, he states:

"a matroid is binary if and only if for each choice of circuits, $C_1,\dots,C_t$, the set $C_1\bigtriangleup\dots\bigtriangleup C_t$ can be partitioned into circuits"

Three questions:

  1. What does the $\bigtriangleup$ operator do?
  2. Is there a more intuitive understanding of what independence sets in a binary matroid are? Is it like a linear matroid, where we have a binary matrix and a subset of $E$ is independent if the corresponding columns of the matrix are linearly independent?
  3. Can we have an independent set of a matroid which has `precedence' conditions? What I mean is that $x \cup U \in \mathcal{I} \iff y \in U$ for some $x,y \in E$, $U \subset E$. I'm pretty sure the answer is no, but since I'm new to this concept I thought I should ask.

Some background on why I'm interested in this: I've managed to pose a set of routing problems for robotics in terms of a submodular maximization problem, and am trying to understand how we can use matroid constraints in this context. It would be very nice if we could represent basic "and" and "or" logical constraints this way, but from what I read I think introducing "and" constraints is fundamentally different from these notions of independence (which makes a lot of sense, considering the greedy algorithm ignores such things and finds the maximal weight basis).

[1] Schrijver, Alexander. Combinatorial optimization: polyhedra and efficiency. Vol. 24. Springer Science & Business Media, 2002. Volume B.

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    Can you give a citation for the particular Schrijver text you're reading? – 2017-01-26

2 Answers 2

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Here is a partial answer:

(2) Yes. A matroid is binary if and only if it is representable over $\mathbb{F}_2$. See, e.g., the wikipedia page on binary matroids.

(3) No. The empty set is an independent set of any (nontrivial) matroid that violates the stated condition. But a similar condition for precedence is captured by the class of accessible set systems, of which matroids are a subclass. See the wikipedia on greedoids for the definition of accessible set system and how to generalize the greedy algorithm for matroids to more general objects.

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    Right - should have tried working out a simpler example. Thank you! – 2017-01-26
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@Aaron-dall answered the second two parts, and in this context $\bigtriangleup$ is the symmetric difference. Mysteries solved.