203 research outputs found
The Dilworth Number of Auto-Chordal-Bipartite Graphs
The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E)
has the same color classes X and Y as B, and two vertices x in X and y in Y are
adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal
bipartite if none of its induced subgraphs is a chordless cycle with at least
six vertices. In this paper, we deal with chordal bipartite graphs whose mirror
is chordal bipartite as well; we call these graphs auto-chordal bipartite
graphs (ACB graphs for short). We describe the relationship to some known graph
classes such as interval and strongly chordal graphs and we present several
characterizations of ACB graphs. We show that ACB graphs have unbounded
Dilworth number, and we characterize ACB graphs with Dilworth number k
On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry
We consider a common type of symmetry where we have a matrix of decision
variables with interchangeable rows and columns. A simple and efficient method
to deal with such row and column symmetry is to post symmetry breaking
constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and
negative results on posting such symmetry breaking constraints. On the positive
side, we prove that we can compute in polynomial time a unique representative
of an equivalence class in a matrix model with row and column symmetry if the
number of rows (or of columns) is bounded and in a number of other special
cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are
often effective in practice, they can leave a large number of symmetric
solutions in the worst case. In addition, we prove that propagating DOUBLELEX
completely is NP-hard. Finally we consider how to break row, column and value
symmetry, correcting a result in the literature about the safeness of combining
different symmetry breaking constraints. We end with the first experimental
study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark
problems.Comment: To appear in the Proceedings of the 16th International Conference on
Principles and Practice of Constraint Programming (CP 2010
An Optimal Algorithm To Recognize Robinsonian Dissimilarities
International audienceA dissimilarity D on a finite set S is said to be Robinsonian if S can be totally ordered in such a way that, for every and . Intuitively, D is Robinsonian if S can be represented by points on a line. Recognizing Robinsonian dissimilarities has many applications in se-riation and classification. In this paper, we present an optimal algorithm to recognize Robinsonian dissimilarities, where n is the cardinal of S. Our result improves the already known algorithms
Box Covers and Domain Orderings for Beyond Worst-Case Join Processing
Recent beyond worst-case optimal join algorithms Minesweeper and its
generalization Tetris have brought the theory of indexing and join processing
together by developing a geometric framework for joins. These algorithms take
as input an index , referred to as a box cover, that stores output
gaps that can be inferred from traditional indexes, such as B+ trees or tries,
on the input relations. The performances of these algorithms highly depend on
the certificate of , which is the smallest subset of gaps in
whose union covers all of the gaps in the output space of a query
. We study how to generate box covers that contain small size certificates
to guarantee efficient runtimes for these algorithms. First, given a query
over a set of relations of size and a fixed set of domain orderings for the
attributes, we give a -time algorithm called GAMB which generates
a box cover for that is guaranteed to contain the smallest size certificate
across any box cover for . Second, we show that finding a domain ordering to
minimize the box cover size and certificate is NP-hard through a reduction from
the 2 consecutive block minimization problem on boolean matrices. Our third
contribution is a -time approximation algorithm called ADORA to
compute domain orderings, under which one can compute a box cover of size
, where is the minimum box cover for under any domain
ordering and is the maximum arity of any relation. This guarantees
certificates of size . We combine ADORA and GAMB with Tetris to
form a new algorithm we call TetrisReordered, which provides several new beyond
worst-case bounds. On infinite families of queries, TetrisReordered's runtimes
are unboundedly better than the bounds stated in prior work
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of
classical domination where selected vertices may have different domination
powers. The minimum cost of a dominating broadcast in a graph is denoted
. The dual of this problem is called multipacking: a multipacking
is a set of vertices such that for any vertex and any positive integer
, the ball of radius around contains at most vertices of .
The maximum size of a multipacking in a graph is denoted mp(G). Naturally
mp(G) . Earlier results by Farber and by Lubiw show that
broadcast and multipacking numbers are equal for strongly chordal graphs. In
this paper, we show that all large grids (height at least 4 and width at least
7), which are far from being chordal, have their broadcast and multipacking
numbers equal
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