1,222 research outputs found
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
We are motivated by a tantalizing open question in exact algorithms: can we
detect whether an -vertex directed graph has a Hamiltonian cycle in time
significantly less than ? We present new randomized algorithms that
improve upon several previous works:
1. We show that for any constant and prime we can count the
Hamiltonian cycles modulo in
expected time less than for a constant that depends only on and
. Such an algorithm was previously known only for the case of counting
modulo two [Bj\"orklund and Husfeldt, FOCS 2013].
2. We show that we can detect a Hamiltonian cycle in
time and polynomial space, where is the size of the maximum
independent set in . In particular, this yields an time
algorithm for bipartite directed graphs, which is faster than the
exponential-space algorithm in [Cygan et al., STOC 2013].
Our algorithms are based on the algebraic combinatorics of "incidence
assignments" that we can capture through evaluation of determinants of
Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed
graphs. In addition to the novel algorithms for directed Hamiltonicity, we use
the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting
out-branchings. Specifically, we give an -time randomized algorithm
for detecting out-branchings with at least internal vertices, improving
upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015].
We also present an algebraic algorithm for the directed -Leaf problem, based
on a non-standard monomial detection problem
Determinant Sums for Undirected Hamiltonicity
We present a Monte Carlo algorithm for Hamiltonicity detection in an
-vertex undirected graph running in time. To the best of
our knowledge, this is the first superpolynomial improvement on the worst case
runtime for the problem since the bound established for TSP almost
fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the
first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard
problems.
For bipartite graphs, we improve the bound to time. Both the
bipartite and the general algorithm can be implemented to use space polynomial
in .
We combine several recently resurrected ideas to get the results. Our main
technical contribution is a new reduction inspired by the algebraic sieving
method for -Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the
Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle
covers over a finite field of characteristic two. We reduce Hamiltonicity to
Labeled Cycle Cover Sum and apply the determinant summation technique for Exact
Set Covers (Bj\"orklund STACS 2010) to evaluate it.Comment: To appear at IEEE FOCS 201
Absolutely Koszul algebras and the Backelin-Roos property
We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos
property and their behavior under standard algebraic operations. In particular,
we identify some Veronese subrings of polynomial rings that have the
Backelin-Roos property and conjecture that the list is indeed complete. Among
other things, we prove that every universally Koszul ring defined by monomials
has the Backelin-Roos property
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