On Complexity Reduction of the LP Bound Computation and Related Problems

A minimal set of shannon-type inequalities for functional dependence structures

“…Though the graph based algorithm does not always give all implied functional dependence relations, it gives many functional dependencies depending on the structure of local functional dependencies without depending on the linear programming framework. In [9] (see also [8]), we gave algorithms to directly obtain a reduced size matrix defining the constraint region. Despite the fact that the reduction was not minimal, it was demonstrated that for the well known butterfly network, the matrix size can be reduced by 98%.…”

“…The number of variables, 2 n − 1, in the optimization problems can be reduced to the number of equivalence classes for a given functional dependency structure. Given the functional dependence structure, the minimal set of inequalities defining the constraint region can also be obtained in a matrix form directly using the approach similar to [9]. Applications of this matrix include solving the optimization problem (10).…”

“…As a continuation of research work in this direction, we aim to employ the results of this paper to develop more refined algorithms (compared to [9]), for obtaining the "minimal" matrix directly. Moreover, we are investigating further generalizations of the ideas presented in this paper.…”

Symmetry in distributed storage systems

“…Example 2 (The LP bound dimensions reduction for (3, 2, 2, α, β)-distributed storage system). The first reduction is obtained using functional dependence relationships among random variables [9], [10]. Below are the maximal irreducible sets [9] for the distributed storage system.…”

Compact representation of polymatroid axioms for random variables with conditional independencies

“…In [1], we considered complexity reduction of the LP bound via simplified characterization of elemental inequalities when network coding and source independence constraints are given for an instance of network coding model. We also gave novel algorithms which directly generate the simplified characterization.…”