On an optimization problem with nested constraints

AbstractWe describe algorithms for solving the integer programming problem maximise ∑j=1n⨍j(xj),subject to ∑jϵSixj⩽bi, i=1,…,m,xj⩾0, j=1,…,n, where the ⨍i are concave nondecreasing and the Si form a nested collection of sets. For the general problem, we present an algorithm of time-complexity O(n log2 n log b), where b is less than the largest of the bi. We also examine the case in which all ⨍i are identical and give an algorithm requiring O(n + m log m) time. Both algorithms use only O(n) space

On the Computational Complexity of Measuring Global Stability of Banking Networks

Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global markets is the so-called financial contagion, namely the spreading of instabilities or failures of individual components of the network to other, perhaps healthier, components. This leads to a natural question of whether the regulatory authorities could have predicted and perhaps mitigated the current economic crisis by effective computations of some stability measure of the banking networks. Motivated by such observations, we consider the problem of defining and evaluating stabilities of both homogeneous and heterogeneous banking networks against propagation of synchronous idiosyncratic shocks given to a subset of banks. We formalize the homogeneous banking network model of Nier et al. and its corresponding heterogeneous version, formalize the synchronous shock propagation procedures, define two appropriate stability measures and investigate the computational complexities of evaluating these measures for various network topologies and parameters of interest. Our results and proofs also shed some light on the properties of topologies and parameters of the network that may lead to higher or lower stabilities.Comment: to appear in Algorithmic

The IntSat method for integer linear programming

Conflict-Driven Clause-Learning (CDCL) SAT solvers can automatically solve very large real-world problems. To go beyond, and in particular in order to solve and optimize problems involving linear arithmetic constraints, here we introduce IntSat, a generalization of CDCL to Integer Linear Programming (ILP). Our simple 1400-line C++ prototype IntSat implementation already shows some competitiveness with commercial solvers such as CPLEX or Gurobi. Here we describe this new IntSat ILP solving method, show how it can be implemented efficiently, and discuss a large list of possible enhancements and extensions.Postprint (author’s final draft