We investigate the preemptive scheduling of periodic, real-time task systems on one processor. First, we show that when all parameters to the system are integers, we may assume without loss of generality that all preemptions occur at integer time values. We then assume, for the remainder of the paper, that all parameters are indeed integers. We then give as our main lemma both necessary and sufficient conditions for a task system to be feasible on one processor. Although these conditions cannot, in general, be tested efficiently (unless P = NP), they do allow us to give efficient algorithms for deciding feasibility on one processor for certain types of periodic task systems. For example, we give a pseudo-polynomial time algorithm for synchronous systems whose densities are bounded by a fixed constant less than 1. This algorithm represents an exponential improvement over the previous best algorithm. We also give a polynomial-time algorithm for systems having a fixed number of distinct typ...

. The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NP-complete, even when all variables are binary. This paper deals with integer linear minimization problems in n variables subject to m linear constraints with at most two variables per inequality, and with all variables bounded between 0 and U . For such systems, a 2\Gammaapproximation algorithm is presented that runs in time O(mnU 2 log(Un 2 =m)), so it is polynomial in the input size if the upper bound U is polynomially bounded. The algorithm works by finding first a super-optimal feasible solution that consists of integer multiples of 1 2 . That solution gives a tight bound on the value of the minimum. It further more has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, ...

We show that a 2-variable integer program defined by m constraints involving coefficients with at most s bits can be solved with O(m+s log m) arithmetic operations or with O(m+logm log s)M(s) bit operations, where M(s) is the time needed for s-bit integer multiplication.

We show that a positive definite integral ternary form can be reduced with O(M(s)log s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication.

This paper offers a brief overview of the developments in combinatorial optimization during the past decade. We discuss improvements in polynomialtime algorithms for problems on graphs and networks, and review the methodological and computational progress in linear and integer optimization. Some of the more prominent software packages in these areas are mentioned. With respect to obtaining approximate solutions to NP-hard problems, we survey recent positive and negative results on polynomial-time approximability and summarize the advances in local search.