Matroid Secretary for Regular and Decomposable Matroids

Abstract

In the matroid secretary problem we are given a stream of elements in random order and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g. graphic, cographic, and transversal matroids). In particular, there is very little known about F -representable matroids (the class of matroids that can be represented as elements of a vector space over a field F ), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)-competitive algorithms for regular matroids (the class of matroids that are representable over any field), and use techniques that are fundamentally matroid-theoretic rather than graphtheoretic.Our main technique is to leverage the seminal regular matroid decomposition theorem of Seymour, which gives a method for decomposing any regular matroid into matroids which are either graphic, cographic, or isomorphic to a simple 10-element matroid. We show how to combine in a black-box manner any algorithms for these basic classes into an algorithm for a given regular matroid, i.e. how to respect the decomposition. In fact, this allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid.

On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

“…Some recent works, solving discrete optimization problems for regular matroids, have used a further stronger version of Seymour's theorem. The strongest form was presented recently by Dinitz and Kortsarz [7], which gives the flexibility to decompose a regular matroid in many different possible sequences. They used it to solve the matroid secretary problem for regular matroids [7].…”

“…The strongest form was presented recently by Dinitz and Kortsarz [7], which gives the flexibility to decompose a regular matroid in many different possible sequences. They used it to solve the matroid secretary problem for regular matroids [7]. Later, Fomin, Golovach, Lokshtanov, and Saurabh utilized it in designing parameterized algorithms for the space cover problem [8] and the spanning circuits problem [9].…”

“…Using this associativity, Seymour's Theorem can be adapted to give, what we call, an unordered decomposition tree (UDT) of a regular matroid M , which allows us to construct many different possible decomposition trees for M . This form of Seymour's Theorem seems to be a folklore knowledge, but is first formally given in [7] (also see [17]). We believe that using this more structured version of Seymour's theorem is necessary to obtain a result corresponding to an arbitrary multiplicative factor α.…”

“…It is possible that the common set S 2 between M 1 and M 3 △M 4 has elements in both M 3 and M 4 . Dinitz and Kortsarz [7] call such a set S 2 as a bad sum-set. More generally, they define a notion of good or bad for a decomposition tree of a regular matroid obtained from Seymour's Theorem.…”

“…Definition 5.4 (Good decomposition tree [7]). The decomposition tree BT(M ) of a regular matroid M , as in Theorem 5.1, is said to be good if for every internal vertex M 0 of the tree BT(M ), which is a k-sum of its two children M 1 and M 2 , the common set S between M 1 and M 2 is completely contained in one of the leaf vertices of the subtree rooted at M 1 and also in one of the leaf vertices of the subtree rooted at M 2 .…”