130 research outputs found
Maximum matching width: new characterizations and a fast algorithm for dominating set
We give alternative definitions for maximum matching width, e.g. a graph
has if and only if it is a subgraph of a chordal
graph and for every maximal clique of there exists with and such that any subset of
that is a minimal separator of is a subset of either or .
Treewidth and branchwidth have alternative definitions through intersections of
subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We
show that mm-width combines both aspects, focusing on nodes and on edges. Based
on this we prove that given a graph and a branch decomposition of mm-width
we can solve Dominating Set in time , thereby beating
whenever . Note that and these inequalities are
tight. Given only the graph and using the best known algorithms to find
decompositions, maximum matching width will be better for solving Dominating
Set whenever
Solving MaxSAT and #SAT on structured CNF formulas
In this paper we propose a structural parameter of CNF formulas and use it to
identify instances of weighted MaxSAT and #SAT that can be solved in polynomial
time. Given a CNF formula we say that a set of clauses is precisely satisfiable
if there is some complete assignment satisfying these clauses only. Let the
ps-value of the formula be the number of precisely satisfiable sets of clauses.
Applying the notion of branch decompositions to CNF formulas and using ps-value
as cut function, we define the ps-width of a formula. For a formula given with
a decomposition of polynomial ps-width we show dynamic programming algorithms
solving weighted MaxSAT and #SAT in polynomial time. Combining with results of
'Belmonte and Vatshelle, Graph classes with structured neighborhoods and
algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get
polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of
structured CNF formulas. For example, we get algorithms for
formulas of clauses and variables and size , if has a linear
ordering of the variables and clauses such that for any variable occurring
in clause , if appears before then any variable between them also
occurs in , and if appears before then occurs also in any clause
between them. Note that the class of incidence graphs of such formulas do not
have bounded clique-width
Hierarchical Clusterings of Unweighted Graphs
We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved
MAP- and MLE-Based Teaching
Imagine a learner L who tries to infer a hidden concept from a collection of
observations. Building on the work [4] of Ferri et al., we assume the learner
to be parameterized by priors P(c) and by c-conditional likelihoods P(z|c)
where c ranges over all concepts in a given class C and z ranges over all
observations in an observation set Z. L is called a MAP-learner (resp. an
MLE-learner) if it thinks of a collection S of observations as a random sample
and returns the concept with the maximum a-posteriori probability (resp. the
concept which maximizes the c-conditional likelihood of S). Depending on
whether L assumes that S is obtained from ordered or unordered sampling resp.
from sampling with or without replacement, we can distinguish four different
sampling modes. Given a target concept c in C, a teacher for a MAP-learner L
aims at finding a smallest collection of observations that causes L to return
c. This approach leads in a natural manner to various notions of a MAP- or
MLE-teaching dimension of a concept class C. Our main results are: We show that
this teaching model has some desirable monotonicity properties. We clarify how
the four sampling modes are related to each other. As for the (important!)
special case, where concepts are subsets of a domain and observations are
0,1-labeled examples, we obtain some additional results. First of all, we
characterize the MAP- and MLE-teaching dimension associated with an optimally
parameterized MAP-learner graph-theoretically. From this central result, some
other ones are easy to derive. It is shown, for instance, that the MLE-teaching
dimension is either equal to the MAP-teaching dimension or exceeds the latter
by 1. It is shown furthermore that these dimensions can be bounded from above
by the so-called antichain number, the VC-dimension and related combinatorial
parameters. Moreover they can be computed in polynomial time
The Perfect Matching Cut Problem Revisited
Under embargo until: 2022-09-20In a graph, a perfect matching cut is an edge cut that is a perfect matching. perfect matching cut (pmc) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP -complete. We revisit the problem and show that pmc remains NP -complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which pmc is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no O∗(2o(n)) -time algorithm for pmc even when restricted to n-vertex bipartite graphs, and also show that pmc can be solved in O∗(1.2721n) time by means of an exact branching algorithm.acceptedVersio
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