53 research outputs found
Treewidth and related graph parameters
For modeling some practical problems, graphs play very important roles.
Since many modeled problems can be NP-hard in general, some restrictions
for inputs are required. Bounding a graph parameter of the inputs is one of
the successful approaches. We study this approach in this thesis. More precisely,
we study two graph parameters, spanning tree congestion and security
number, that are related to treewidth.
Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T),
the congestion of e is the number of edges in G connecting two components
of T − e. The edge congestion of G in T is the maximum congestion over all
edges in T. The spanning tree congestion of G is the minimum congestion
of G in its spanning trees. In this thesis, we show the spanning tree congestion
for the complete k-partite graphs, the two-dimensional tori, and the twodimensional
Hamming graphs. We also address lower bounds of spanning
tree congestion for the multi-dimensional hypercubes, the multi-dimensional
grids, and the multi-dimensional Hamming graphs.
The security number of a graph is the cardinality of a smallest vertex subset
of the graph such that any “attack” on the subset is “defendable.” In this thesis,
we determine the security number of two-dimensional cylinders and tori.
This result settles a conjecture of Brigham, Dutton and Hedetniemi [Discrete
Appl. Math. 155 (2007) 1708–1714]. We also show that every outerplanar
graph has security number at most three. Additionally, we present lower and
upper bounds for some classes of graphs.学位記番号:工博甲39
Weak Coloring Numbers of Intersection Graphs
Weak and strong coloring numbers are generalizations of the degeneracy of a
graph, where for each natural number , we seek a vertex ordering such every
vertex can (weakly respectively strongly) reach in steps only few vertices
with lower index in the ordering. Both notions capture the sparsity of a graph
or a graph class, and have interesting applications in the structural and
algorithmic graph theory. Recently, the first author together with McCarty and
Norin observed a natural volume-based upper bound for the strong coloring
numbers of intersection graphs of well-behaved objects in , such
as homothets of a centrally symmetric compact convex object, or comparable
axis-aligned boxes.
In this paper, we prove upper and lower bounds for the -th weak coloring
numbers of these classes of intersection graphs. As a consequence, we describe
a natural graph class whose strong coloring numbers are polynomial in , but
the weak coloring numbers are exponential. We also observe a surprising
difference in terms of the dependence of the weak coloring numbers on the
dimension between touching graphs of balls (single-exponential) and hypercubes
(double-exponential)
New Results for the MAP Problem in Bayesian Networks
This paper presents new results for the (partial) maximum a posteriori (MAP)
problem in Bayesian networks, which is the problem of querying the most
probable state configuration of some of the network variables given evidence.
First, it is demonstrated that the problem remains hard even in networks with
very simple topology, such as binary polytrees and simple trees (including the
Naive Bayes structure). Such proofs extend previous complexity results for the
problem. Inapproximability results are also derived in the case of trees if the
number of states per variable is not bounded. Although the problem is shown to
be hard and inapproximable even in very simple scenarios, a new exact algorithm
is described that is empirically fast in networks of bounded treewidth and
bounded number of states per variable. The same algorithm is used as basis of a
Fully Polynomial Time Approximation Scheme for MAP under such assumptions.
Approximation schemes were generally thought to be impossible for this problem,
but we show otherwise for classes of networks that are important in practice.
The algorithms are extensively tested using some well-known networks as well as
random generated cases to show their effectiveness.Comment: A couple of typos were fixed, as well as the notation in part of
section 4, which was misleading. Theoretical and empirical results have not
change
Recontamination Helps a Lot to Hunt a Rabbit
The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number h(G) of a graph G is the minimum integer k such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes...), but the computational complexity of computing h(G) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot "must not host the rabbit anymore". This allows us to obtain new results in various graph classes.
More precisely, let the monotone hunter number mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ? mh(G) ? pw(G)+1 for any graph G with pathwidth pw(G), which implies that computing mh(G), or even approximating mh(G) up to an additive constant, is NP-hard. Then, we show that mh(G) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, we show that, for every k ? 3, there exists a tree T with h(T) = 2 and mh(T) = k. We conclude by proving that computing h (resp., mh) is FPT parameterised by the minimum size of a vertex cover
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