8 research outputs found
Exact Covers via Determinants
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such
that every hyperedge includes one vertex from each part, the k-dimensional
matching problem asks whether there is a disjoint collection of the hyperedges
which covers all vertices. We show it can be solved by a randomized polynomial
space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors
polynomial in n and k.
When we drop the partition constraint and permit arbitrary hyperedges of
cardinality k, we obtain the exact cover by k-sets problem. We show it can be
solved by a randomized polynomial space algorithm in time O*(c_k^n), where
c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k.
Both results substantially improve on the previous best algorithms for these
problems, especially for small k, and follow from the new observation that
Lovasz' perfect matching detection via determinants (1979) admits an embedding
in the recently proposed inclusion-exclusion counting scheme for set covers,
despite its inability to count the perfect matchings
Exact Covers via Determinants
Given a -uniform hypergraph on vertices, partitioned in equal parts such that every hyperedge includes one vertex from each part, the -Dimensional Matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices.
We show it can be solved by a randomized polynomial space algorithm in time. The notation hides factors
polynomial in and .
The general Exact Cover by -Sets problem asks the same when the partition constraint is dropped and arbitrary hyperedges of cardinality are permitted. We show it can be solved by a randomized polynomial space algorithm in time, where , and provide a general bound for larger .
Both results substantially improve on the previous best algorithms for these problems, especially for small . They follow from the new observation that Lov\u27asz\u27 perfect matching detection via determinants (Lov\u27asz, 1979) admits an embedding in the recently proposed inclusion--exclusion counting scheme for set covers, emph{despite} its inability to count the perfect matchings
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
Narrow sieves for parameterized paths and packings
We present randomized algorithms for some well-studied, hard combinatorial
problems: the k-path problem, the p-packing of q-sets problem, and the
q-dimensional p-matching problem. Our algorithms solve these problems with high
probability in time exponential only in the parameter (k, p, q) and using
polynomial space; the constant bases of the exponentials are significantly
smaller than in previous works. For example, for the k-path problem the
improvement is from 2 to 1.66. We also show how to detect if a d-regular graph
admits an edge coloring with colors in time within a polynomial factor of
O(2^{(d-1)n/2}).
Our techniques build upon and generalize some recently published ideas by I.
Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010,
FOCS 2010)
The Asymptotic Rank Conjecture and the Set Cover Conjecture are not Both True
Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a
strong submultiplicative upper bound on the rank of a three-tensor obtained as
an iterated Kronecker product of a constant-size base tensor. The conjecture,
if true, most notably would put square matrix multiplication in quadratic time.
We note here that some more-or-less unexpected algorithmic results in the area
of exponential-time algorithms would also follow. Specifically, we study the
so-called set cover conjecture, which states that for any there
exists a positive integer constant such that no algorithm solves the
-Set Cover problem in worst-case time . The -Set Cover problem asks, given as input an
-element universe , a family of size-at-most- subsets of
, and a positive integer , whether there is a subfamily of at most
sets in whose union is . The conjecture was formulated by Cygan
et al. in the monograph Parameterized Algorithms [Springer, 2015] but was
implicit as a hypothesis already in Cygan et al. [CCC 2012, ACM Trans.
Algorithms 2016], there conjectured to follow from the Strong Exponential Time
Hypothesis. We prove that if the asymptotic rank conjecture is true, then the
set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi
[STACS 2019], in this scenario we would also get a
-time randomized algorithm for some constant
for another well-studied problem for which no such algorithm is
known, namely that of deciding whether a given -vertex directed graph has a
Hamiltonian cycle