Matroid Secretary for Regular and Decomposable Matroids
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 irrevocab...
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| Vydáno v: | SIAM journal on computing Ročník 43; číslo 5; s. 1807 - 1830 |
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| Jazyk: | angličtina |
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01.01.2014
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| ISSN: | 0097-5397, 1095-7111 |
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| 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 [Proceedings of SODA' 07] 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 graph-theoretic. 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 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 ofmax-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid. |
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| AbstractList | 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 [Proceedings of SODA' 07] 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 graph-theoretic. 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 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 ofmax-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid. |
| Author | Kortsarz, Guy Dinitz, Michael |
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| References | Jaillet P. (atypb15) 2013 Dynkin E. B. (atypb8) 1963 Lovász L. (atypb19) 2006; 43 Korula N. (atypb17) 2009 atypb26 atypb11 atypb22 Feldman M. (atypb9) 2011 atypb23 Gupta A. (atypb13) 2010 Tutte W. T. (atypb28) 1958; 88 atypb2 Gharan S. O. (atypb12) 2011 atypb4 atypb7 atypb6 |
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| Title | Matroid Secretary for Regular and Decomposable Matroids |
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