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**1 - 7**of**7**### Matching, Matroids, and Extensions

, 2001

"... Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, tw ..."

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Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones -- jump systems and path-matchings.

### Counting Bases

"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my readers. I understand that my thesis may be made electronically available to the public. ii A theorem of Edmonds characterizes when a pair of matroi ..."

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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my readers. I understand that my thesis may be made electronically available to the public. ii A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #P-complete problem of counting the number of perfect matchings in a bipartite graph. We focus on the problem of counting the common bases in pairs of regular matroids, and describe a class called Pfaffian matroid pairs for which this enumeration problem can be solved. We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids. When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-

### Abstract The Complexity of Matrix Completion

"... Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applic ..."

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Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applications. Determining the complexity of this problem is a fundamental open question in computational complexity. Under different settings of parameters, the problem is variously in P, in RP, or NP-hard. We shed new light on this landscape by demonstrating a new region of NP-hard scenarios. As a special case, we obtain the first known hardness result for matrices in which each variable appears only twice. Another particular scenario that we consider is the simultaneous matrix completion problem, where one must simultaneously maximize the rank for several matrices that share variables. This problem has important applications in the field of network coding. Recent work has given a simple, greedy, deterministic algorithm for this problem, assuming that the algorithm works over a sufficiently large field. We show an exact threshold for the field size required to find a simultaneous completion efficiently. This result implies that, surprisingly, the simple greedy algorithm is optimal: finding a simultaneous completion over any smaller field is NPhard. 1

### An Algebraic Approach to Matching Problems

"... In Tutte's seminal paper on matching, he associates a skew{symmetric matrix with a graph; this matrix is now known as the Tutte matrix. The rank of the Tutte matrix is exactly twice the size of a maximum matching in the graph. This formulation easily leads to an ecient randomized algorithm f ..."

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In Tutte's seminal paper on matching, he associates a skew{symmetric matrix with a graph; this matrix is now known as the Tutte matrix. The rank of the Tutte matrix is exactly twice the size of a maximum matching in the graph. This formulation easily leads to an ecient randomized algorithm for matching. The Tutte matrix is also useful in obtaining a min{max theorem and an ecient deterministic algorithm. We review these results and look at similar formulations of other problems; namely, linear matroid intersection, linear matroid parity, path matching, and matching forests.

### Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem

"... We give an algebraic, determinant-based algorithm for the K-Cycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the O ∗ (2 |K | ) running time of the algorithm of Björklund et al. (SODA, 2012). F ..."

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We give an algebraic, determinant-based algorithm for the K-Cycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the O ∗ (2 |K | ) running time of the algorithm of Björklund et al. (SODA, 2012). Furthermore, our approach is open for treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to a polynomial compression of the problem, i.e., a polynomial-time reduction of the K-Cycle problem into an algebraic problem with coding size O(|K | 3). This is surprising, as several related problems (e.g., k-Cycle and the Disjoint Paths problem) are known not to admit such a reduction unless the polynomial hierarchy collapses. Furthermore, despite the result, we are not aware of any witness for the K-Cycle problem of size polynomial in |K | + log n, which seems (for now) to separate the notions of polynomial compression and polynomial kernelization (as a polynomial kernelization for a problem in NP necessarily implies a small witness).