A perfect matching of a complete graph K 2n is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if F is family of intersecting perfect matchings of K 2n , then |F | ≤ (2(n − 1) − 1)!! and if equality holds, then F = F ij where F ij is the family of all perfect matchings of K 2n that contain some fixed edge ij. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family F is non-Hamiltonian, that is, m ∪ m ′ ∼ = C 2n for any m, m ′ ∈ F , then |F | ≤ (2(n − 1) − 1)!! and this bound is met with equality if and only if F = F ij . Our results make ample use of a somewhat understudied symmetric commutative association scheme arising from the Gelfand pair (S 2n , S 2 ≀ Sn). We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.
For even k ∈ N, the matchings connectivity matrix M k is a binary matrix indexed by perfect matchings on k vertices; the entry at (M, M ) is 1 iff M ∪ M forms a single cycle. Cygan et al. (STOC 2013) showed that the rank of M k over Z 2 is Θ( √ 2 k ) and used this to give an O * ((2 + √ 2) pw ) time algorithm for counting Hamiltonian cycles modulo 2 on graphs of pathwidth pw, carrying over to the decision problem via witness isolation. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within M k , which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).We present a new technique for a similar "pattern propagation" when only a black-box lower bound on the asymptotic rank of M k is given; no stronger structural insights such as the existence of large permutation submatrices in M k are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes p) parameterized by pathwidth.To apply this technique, we prove that the rank of M k over the rationals is 4 k /poly(k), using the representation theory of the symmetric group and various insights from algebraic combinatorics. We also show that the rank of M k over Z p is Ω(1.57 k ) for any prime p = 2. IntroductionRank is a fundamental concept in linear algebra and has numerous applications in diverse areas of discrete mathematics and theoretical computer science, such as algebraic complexity [8], communication complexity [27], and extremal combinatorics [30], to name only a few examples. A common phenomenon in these areas is that low rank often helps in proving combinatorial upper bounds or designing algorithms, e.g., through representative sets [7,13,21] or the polynomial method (which ultimately relies on fast rectangular matrix multiplication, enabled through low-rank factorizations of problem-related matrices [37]). In particular, rank has recently found applications in fine-grained complexity (see [1] and the references therein) and the closely related area of parameterized complexity. In the latter, several influential results, such as algorithms for kernelization [21], the longest path problem [31], and connectivity problems parameterized by treewidth [12,11,7], rely crucially on low-rank factorizations.In view of the utility of low rank in proving upper bounds, it is natural to ask, conversely, whether high rank translates into lower bounds. Indeed, examples for this connection can be found in communication complexity [22, Section 1.4] and circuit complexity [16]. In the present paper, we find such applications also in finegrained and parameterized complexity: We develop a technique that allows us to transform rank lower bounds into conditional lower bounds for the problem #HC of counting Hamiltonian cycles. The decision version HC of #HC, which asks for the existenc...
A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph G is a digraph obtained from a sequence of vertex complement operations on G. Dahlhaus et al. showed that, given an adjacency-list representation of G, depth-first search (DFS) on G can be performed in O(n + m) time, where n is the number of vertices and m is the number of edges in G. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive O(n + m) algorithm that uses no complicated data-structures.
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a lineartime algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a lineartime algorithm to find one in any binary matrix that does not have the consecutive-ones property.
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