31,795 research outputs found
Linear-Delay Enumeration for Minimal Steiner Problems
Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv,
Inf. Syst. 2008] pointed out the problem of enumerating -fragments is of
great importance in a keyword search on data graphs. In a graph-theoretic term,
the problem corresponds to enumerating minimal Steiner trees in (directed)
graphs. In this paper, we propose a linear-delay and polynomial-space algorithm
for enumerating all minimal Steiner trees, improving on a previous result in
[Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be
extended to other Steiner problems, such as minimal Steiner forests, minimal
terminal Steiner trees, and minimal directed Steiner trees. As another variant
of the minimal Steiner tree enumeration problem, we study the problem of
enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay
and exponential-space enumeration algorithm of minimal induced Steiner
subgraphs on claw-free graphs. Contrary to these tractable results, we show
that the problem of enumerating minimal group Steiner trees is at least as hard
as the minimal transversal enumeration problem on hypergraphs
Parameterized Study of Steiner Tree on Unit Disk Graphs
We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R? V(G) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V? R. The vertices of R are referred to as terminals and the vertices of V(G)? R as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in n^{O(?{t+k})} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2^{O(k)}n^{O(1)}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs [Fomin et al., 2019]. We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree on disk graphs parameterized by k is W[1]-hard
Further improvements of Steiner tree approximations
The Steiner tree problem requires to find a shortest tree connecting a given set of terminal points in a metric space. We suggest a better and fast heuristic for the Steiner problem in graphs and in rectilinear plane. This heuristic finds a Steiner tree at most 1.757 and 1.267 times longer than the optimal solution in graphs and rectilinear plane, respectively
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
We study Steiner Forest on -subgraph-free graphs, that is, graphs that do
not contain some fixed graph as a (not necessarily induced) subgraph. We
are motivated by a recent framework that completely characterizes the
complexity of many problems on -subgraph-free graphs. However, in contrast
to e.g. the related Steiner Tree problem, Steiner Forest falls outside this
framework. Hence, the complexity of Steiner Forest on -subgraph-free graphs
remained tantalizingly open. In this paper, we make significant progress
towards determining the complexity of Steiner Forest on -subgraph-free
graphs. Our main results are four novel polynomial-time algorithms for
different excluded graphs that are central to further understand its
complexity. Along the way, we study the complexity of Steiner Forest for graphs
with a small -deletion set, that is, a small set of vertices such that
each component of has size at most . Using this parameter, we give two
noteworthy algorithms that we later employ as subroutines. First, we prove
Steiner Forest is FPT parameterized by when (i.e. the vertex cover
number). Second, we prove Steiner Forest is polynomial-time solvable for graphs
with a 2-deletion set of size at most 2. The latter result is tight, as the
problem is NP-complete for graphs with a 3-deletion set of size 2
Hardness and Approximation of Octilinear Steiner Trees
Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or 45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a graph of size O(n^2/epsilon^2) which contains a (1+epsilon)-approximation of a minimum octilinear Steiner tree for every epsilon > 0 and n = |K|. Hence, we can apply any k-approximation algorithm for the Steiner tree problem in graphs (the currently best known bound is k=1.55) and achieve an (k+epsilon)-approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons)
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