252 research outputs found
Parameterized Approximation Algorithms for Bidirected Steiner Network Problems
The Directed Steiner Network (DSN) problem takes as input a directed
edge-weighted graph and a set of
demand pairs. The aim is to compute the cheapest network for
which there is an path for each . It is known
that this problem is notoriously hard as there is no
-approximation algorithm under Gap-ETH, even when parametrizing
the runtime by [Dinur & Manurangsi, ITCS 2018]. In light of this, we
systematically study several special cases of DSN and determine their
parameterized approximability for the parameter .
For the bi-DSN problem, the aim is to compute a planar
optimum solution in a bidirected graph , i.e., for every edge
of the reverse edge exists and has the same weight. This problem
is a generalization of several well-studied special cases. Our main result is
that this problem admits a parameterized approximation scheme (PAS) for . We
also prove that our result is tight in the sense that (a) the runtime of our
PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists
for any generalization of bi-DSN, unless FPT=W[1].
One important special case of DSN is the Strongly Connected Steiner Subgraph
(SCSS) problem, for which the solution network needs to strongly
connect a given set of terminals. It has been observed before that for SCSS
a parameterized -approximation exists when parameterized by [Chitnis et
al., IPEC 2013]. We give a tight inapproximability result by showing that for
no parameterized -approximation algorithm exists under
Gap-ETH. Additionally we show that when restricting the input of SCSS to
bidirected graphs, the problem remains NP-hard but becomes FPT for
Forbidden Directed Minors and Kelly-width
Partial 1-trees are undirected graphs of treewidth at most one. Similarly,
partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known
that an undirected graph is a partial 1-tree if and only if it has no K_3
minor. In this paper, we generalize this characterization to partial 1-DAGs. We
show that partial 1-DAGs are characterized by three forbidden directed minors,
K_3, N_4 and M_5
Flows on Bidirected Graphs
The study of nowhere-zero flows began with a key observation of Tutte that in
planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of
k-tensions). Tutte conjectured that every graph without a cut-edge has a
nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero
6-flow.
For a graph embedded in an orientable surface of higher genus, flows are not
dual to colourings, but to local-tensions. By Seymour's theorem, every graph on
an orientable surface without the obvious obstruction has a nowhere-zero
6-local-tension. Bouchet conjectured that the same should hold true on
non-orientable surfaces. Equivalently, Bouchet conjectured that every
bidirected graph with a nowhere-zero -flow has a nowhere-zero
6-flow. Our main result establishes that every such graph has a nowhere-zero
12-flow.Comment: 24 pages, 2 figure
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs
By a T-star we mean a complete bipartite graph K_{1,t} for some t <= T. For an undirected graph G, a T-star packing is a collection of node-disjoint T-stars in G.
For example, we get ordinary matchings for and packings of paths of length 1 and 2 for . Hereinafter we assume that T >= 2.
Hell and Kirkpatrick devised an ad-hoc augmenting algorithm that finds a T-star packing covering the maximum number of nodes. The latter algorithm also yields a min-max formula.
We show that T-star packings are reducible to network flows, hence the above problem is solvable in time (hereinafter n denotes the number of nodes in G, and m --- the number of edges).
For the edge-weighted case (in which weights may be assumed positive) finding a maximum -packing is NP-hard. A novel 9/4 T/(T + 1)-factor approximation algorithm is presented.
For non-negative node weights the problem reduces to a special case of a max-cost flow. We develop a divide-and-conquer approach that solves it in O(m sqrt(n) log(n)) time. The node-weighted problem with arbitrary weights is more difficult. We prove that it is NP-hard for T >= 3 and is solvable in strongly-polynomial time for T = 2
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