263 research outputs found
A note on "optimal resource allocation for security in reliability systems"
In a recent paper by Azaiez and Bier [Azaiez, M.N., Bier, V.M., 2007. Optimal resource allocation for security in reliability systems. European Journal of Operational Research 181, 773–786], the problem of determining resource allocation in series-parallel systems (SPSs) is considered. The results for this problem are based on the results for the least-expected cost failure-state diagnosis problem. In this note, it is demonstrated that the results for the least-expected cost failure-state diagnosis problem for SPSs in Azaiez and Bier (2007) are incorrect. In addition relevant results that were not cited in the paper are summarized
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
Quadratization of Symmetric Pseudo-Boolean Functions
A pseudo-Boolean function is a real-valued function
of binary variables; that is, a mapping from
to . For a pseudo-Boolean function on
, we say that is a quadratization of if is a
quadratic polynomial depending on and on auxiliary binary variables
such that for
all . By means of quadratizations, minimization of is
reduced to minimization (over its extended set of variables) of the quadratic
function . This is of some practical interest because minimization of
quadratic functions has been thoroughly studied for the last few decades, and
much progress has been made in solving such problems exactly or heuristically.
A related paper \cite{ABCG} initiated a systematic study of the minimum number
of auxiliary -variables required in a quadratization of an arbitrary
function (a natural question, since the complexity of minimizing the
quadratic function depends, among other factors, on the number of
binary variables). In this paper, we determine more precisely the number of
auxiliary variables required by quadratizations of symmetric pseudo-Boolean
functions , those functions whose value depends only on the Hamming
weight of the input (the number of variables equal to ).Comment: 17 page
A Potential Reduction Algorithm for Two-person Zero-sum Mean Payoff Stochastic Games
We suggest a new algorithm for two-person zero-sum undiscounted stochastic
games focusing on stationary strategies. Given a positive real , let
us call a stochastic game -ergodic, if its values from any two
initial positions differ by at most . The proposed new algorithm
outputs for every in finite time either a pair of stationary
strategies for the two players guaranteeing that the values from any initial
positions are within an -range, or identifies two initial positions
and and corresponding stationary strategies for the players proving
that the game values starting from and are at least
apart. In particular, the above result shows that if a stochastic game is
-ergodic, then there are stationary strategies for the players
proving -ergodicity. This result strengthens and provides a
constructive version of an existential result by Vrieze (1980) claiming that if
a stochastic game is -ergodic, then there are -optimal stationary
strategies for every . The suggested algorithm is based on a
potential transformation technique that changes the range of local values at
all positions without changing the normal form of the game
- …