29,154 research outputs found
Space Saving by Dynamic Algebraization
Dynamic programming is widely used for exact computations based on tree
decompositions of graphs. However, the space complexity is usually exponential
in the treewidth. We study the problem of designing efficient dynamic
programming algorithm based on tree decompositions in polynomial space. We show
how to construct a tree decomposition and extend the algebraic techniques of
Lokshtanov and Nederlof such that the dynamic programming algorithm runs in
time , where is the maximum number of vertices in the union of
bags on the root to leaf paths on a given tree decomposition, which is a
parameter closely related to the tree-depth of a graph. We apply our algorithm
to the problem of counting perfect matchings on grids and show that it
outperforms other polynomial-space solutions. We also apply the algorithm to
other set covering and partitioning problems.Comment: 14 pages, 1 figur
Below All Subsets for Some Permutational Counting Problems
We show that the two problems of computing the permanent of an
matrix of -bit integers and counting the number of
Hamiltonian cycles in a directed -vertex multigraph with
edges can be reduced to relatively
few smaller instances of themselves. In effect we derive the first
deterministic algorithms for these two problems that run in time in
the worst case. Classic time algorithms for the two
problems have been known since the early 1960's. Our algorithms run in
time.Comment: Corrected several technical errors, added comment on how to use the
algorithm for ATSP, and changed title slightly to a more adequate on
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
Tight Euler tours in uniform hypergraphs - computational aspects
By a tight tour in a -uniform hypergraph we mean any sequence of its
vertices such that for all the set
is an edge of (where operations on
indices are computed modulo ) and the sets for are
pairwise different. A tight tour in is a tight Euler tour if it contains
all edges of . We prove that the problem of deciding if a given -uniform
hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved
in time (where is the number of edges in the input hypergraph),
unless the ETH fails. We also present an exact exponential algorithm for the
problem, whose time complexity matches this lower bound, and the space
complexity is polynomial. In fact, this algorithm solves a more general problem
of computing the number of tight Euler tours in a given uniform hypergraph
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
Dynamic programming based algorithms for set multicover and multiset multicover problems
Given a universe N containing n elements and a collection of multisets or sets over N, the multiset multicover (MSMC) problem or the set multicover (SMC) problem is to cover all elements at least a number of times as specified in their coverage requirements with the minimum number of multisets or sets. In this paper, we give various exact algorithms for these two problems with or without constraints on the number of times a multiset or set may be chosen. First, we show that the MSMC without multiplicity constraints problem can be solved in O* ((b + 1)n | F |) time and polynomial space, where b is the maximum coverage requirement and | F | denotes the total number of given multisets over N. (The O* notation suppresses a factor polynomial in n.) To our knowledge, this is the first known exact algorithm for the MSMC without multiplicity constraints problem. Second, by combining dynamic programming and the inclusion-exclusion principle, we can exactly solve the SMC without multiplicity constraints problem in O ((b + 2)n) time. Compared with two recent results, in [Q.-S. Hua, Y. Wang, D. Yu, F.C.M. Lau, Set multi-covering via inclusion-exclusion, Theoretical Computer Science, 410 (38-40) (2009) 3882-3892] and [J. Nederlof, Inclusion exclusion for hard problems, Master Thesis, Utrecht University, The Netherlands, 2008], respectively, ours is the fastest exact algorithm for the SMC without multiplicity constraints problem. Finally, by directly using dynamic programming, we give the first known exact algorithm for the MSMC or the SMC with multiplicity constraints problem in O ((b + 1)n | F |) time and O* ((b + 1)n) space. This algorithm can also be easily adapted as a constructive algorithm for the MSMC without multiplicity constraints problem. © 2010 Elsevier B.V. All rights reserved.postprin
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