24,175 research outputs found
Guessing numbers based on network routing capacity
In this thesis, a new method of studying problems in network information flow is introduced. This is established by developing on the relatively new concept of guessing numbers, which is related to network coding, and combining it with another technique known as fractional routing. The resultant guessing game, called a fractional guessing game, is a modification of the existing guessing game that incorporates the notion of fractional routing for multiple-unicast networks. It is essentially an intersection of two major areas of mathematics which allows further study into network flow problems, and the relationship between guessing numbers and the solvability of network flow problems. To illustrate this relationship, we construct a graph (the pentagon or 5-cycle) that has a non- integer guessing number, by using a guessing strategy which ensures a guessing number in the half-open interval (2, 2.5]. This is an application of the link between guessing numbers on graphs and the solvability of their related networks. It is shown that the fractional guessing number of an example graph is higher than the guessing number computed using the existing linear approach. This guessing number is also dependent on the size of the alphabet from which the messages are constructed, answering an open question from the literature relating to the possibility of such a dependency. Finally, the extension of the method to include techniques of both routing capacity and network coding is discussed
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
Better abstractions for timed automata
We consider the reachability problem for timed automata. A standard solution
to this problem involves computing a search tree whose nodes are abstractions
of zones. These abstractions preserve underlying simulation relations on the
state space of the automaton. For both effectiveness and efficiency reasons,
they are parametrized by the maximal lower and upper bounds (LU-bounds)
occurring in the guards of the automaton. We consider the aLU abstraction
defined by Behrmann et al. Since this abstraction can potentially yield
non-convex sets, it has not been used in implementations. We prove that aLU
abstraction is the biggest abstraction with respect to LU-bounds that is sound
and complete for reachability. We also provide an efficient technique to use
the aLU abstraction to solve the reachability problem.Comment: Extended version of LICS 2012 paper (conference paper till v6). in
Information and Computation, available online 27 July 201
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