6 research outputs found
Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary
We study the classical Node-Disjoint Paths (NDP) problem: given an undirected n-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set {P} of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy O(sqrt{n})-approximation algorithm. Until recently, the best negative result was an Omega(log^{1/2-epsilon}n)-hardness of approximation, for any fixed epsilon, under standard complexity assumptions.
A special case of the problem, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for this special case achieves an O~(n^{1/4})-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor 2^{Omega(sqrt{log n})}, even if the underlying graph is a subgraph of a grid, and all source vertices lie on the grid boundary. In a very recent follow-up work, the authors further show that NDP in grid graphs is hard to approximate to within factor Omega(2^{log^{1-epsilon}n}) for any constant epsilon under standard complexity assumptions, and to within factor n^{Omega(1/(log log n)^2)} under randomized ETH.
In this paper we study the NDP problem in grid graphs, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized 2^{O(sqrt{log n}* log log n)}-approximation algorithm for this problem. Our result in a sense complements the 2^{Omega(sqrt{log n})}-hardness of approximation for sub-graphs of grids with sources lying on the grid boundary, and should be contrasted with the above-mentioned almost polynomial hardness of approximation of NDP in grid graphs (where the sources and the destinations may lie anywhere in the grid).
Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an Omega(sqrt n) integrality gap, even in grid graphs, with all source and destination vertices lying on the grid boundary. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods
Grid Recognition: Classical and Parameterized Computational Perspectives
Grid graphs, and, more generally, grid graphs, form one of the
most basic classes of geometric graphs. Over the past few decades, a large body
of works studied the (in)tractability of various computational problems on grid
graphs, which often yield substantially faster algorithms than general graphs.
Unfortunately, the recognition of a grid graph is particularly hard -- it was
shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this
paper, we provide several positive results in this regard in the framework of
parameterized complexity (additionally, we present new and complementary
hardness results). Specifically, our contribution is threefold. First, we show
that the problem is fixed-parameter tractable (FPT) parameterized by where is the maximum size of a connected component of
. This also implies that the problem is FPT parameterized by
where is the treedepth of (to be compared with the hardness
for pathwidth 2 where ). Further, we derive as a corollary that strip
packing is FPT with respect to the height of the strip plus the maximum of the
dimensions of the packed rectangles, which was previously only known to be in
XP. Second, we present a new parameterization, denoted , relating graph
distance to geometric distance, which may be of independent interest. We show
that the problem is para-NP-hard parameterized by , but FPT parameterized
by on trees, as well as FPT parameterized by . Third, we show that
the recognition of grid graphs is NP-hard on graphs of pathwidth 2
where . Moreover, when and are unrestricted, we show that the
problem is NP-hard on trees of pathwidth 2, but trivially solvable in
polynomial time on graphs of pathwidth 1
Shortest k-Disjoint Paths via Determinants
The well-known -disjoint path problem (-DPP) asks for pairwise
vertex-disjoint paths between specified pairs of vertices in a
given graph, if they exist. The decision version of the shortest -DPP asks
for the length of the shortest (in terms of total length) such paths. Similarly
the search and counting versions ask for one such and the number of such
shortest set of paths, respectively.
We restrict attention to the shortest -DPP instances on undirected planar
graphs where all sources and sinks lie on a single face or on a pair of faces.
We provide efficient sequential and parallel algorithms for the search versions
of the problem answering one of the main open questions raised by Colin de
Verdiere and Schrijver for the general one-face problem. We do so by providing
a randomised algorithm along with an time randomised
sequential algorithm. We also obtain deterministic algorithms with similar
resource bounds for the counting and search versions.
In contrast, previously, only the sequential complexity of decision and
search versions of the "well-ordered" case has been studied. For the one-face
case, sequential versions of our routines have better running times for
constantly many terminals. In addition, the earlier best known sequential
algorithms (e.g. Borradaile et al.) were randomised while ours are also
deterministic.
The algorithms are based on a bijection between a shortest -tuple of
disjoint paths in the given graph and cycle covers in a related digraph. This
allows us to non-trivially modify established techniques relating counting
cycle covers to the determinant. We further need to do a controlled
inclusion-exclusion to produce a polynomial sum of determinants such that all
"bad" cycle covers cancel out in the sum allowing us to count "good" cycle
covers.Comment: 17 pages, 6 figure