19 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
Faster Detours in Undirected Graphs
The -Detour problem is a basic path-finding problem: given a graph on
vertices, with specified nodes and , and a positive integer , the
goal is to determine if has an -path of length exactly , where is the length of a shortest path from to
. The -Detour problem is NP-hard when is part of the input, so
researchers have sought efficient parameterized algorithms for this task,
running in time, for as slow-growing as possible.
We present faster algorithms for -Detour in undirected graphs, running in
randomized and deterministic
time. The previous fastest algorithms for this problem took randomized and deterministic time
[Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact
that detecting a path of a given length in an undirected graph is easier if we
are promised that the path belongs to what we call a "bipartitioned" subgraph,
where the nodes are split into two parts and the path must satisfy constraints
on those parts. Previously, this idea was used to obtain the fastest known
algorithm for finding paths of length in undirected graphs
[Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017].
Our work has direct implications for the -Longest Detour problem: in this
problem, we are given the same input as in -Detour, but are now tasked with
determining if has an -path of length at least
Our results for k-Detour imply that we can solve -Longest Detour in randomized and deterministic time.
The previous fastest algorithms for this problem took
randomized and deterministic time [Fomin et al.,
STACS 2022]