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**1 - 3**of**3**### Matching Theory

"... y interested in the maximum matching problem; that is, th problem of nding a matching of maximum cardinality. For simplicity, we refer to a matching of maximum cardinality as a maximum matching, and let (G) denote the size of a maximum matching in G. Let M be a matching in G, and let v be a vertex ..."

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y interested in the maximum matching problem; that is, th problem of nding a matching of maximum cardinality. For simplicity, we refer to a matching of maximum cardinality as a maximum matching, and let (G) denote the size of a maximum matching in G. Let M be a matching in G, and let v be a vertex of G. If v is the end of an edge in M , then we say that M saturates v. The set of all vertices saturated by a particular matching is called a matchable set of G. Note that, since each edge saturates two vertices, matchable sets have even cardinality. A matching that saturates every vertex is called perfect. The perfect matching problem is the problem of deciding whether a graph has a perfect matching. Obviously, G

### 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).