14 research outputs found
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
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page