Jiri Sgall

On-line Scheduling

We have seen a variety of on-line scheduling problems. Many of them are understood satisfactorily, but there are also many interesting open problems. Studied scheduling problems differ not only in the setting and numerical results, but also in the techniques used. In this way on-line scheduling illustrates many general aspects of competitive analysis.

Multiprocessor Scheduling with Rejection

We consider a version of multiprocessor scheduling with the special feature that jobs may be rejected at a certain penalty. An instance of the problem is given by m identical parallel machines and a set of n jobs, with each job characterized by a processing time and a penalty. In the on-line version the jobs become available one by one and we have to schedule or reject a job before we have any information about future jobs. The objective is to minimize the makespan of the schedule for accepted jobs plus the sum of the penalties of rejected jobs. The main result is a 1 + 2:618 competitive algorithm for the on-line version of the problem, where is the golden ratio. A matching lower bound shows that this is the best possible algorithm working for all m. For xed m we give improved bounds; in particular, for m = 2 we give a 1:618 competitive algorithm, which is best possible. For the o-line problem we present a fully polynomial approximation scheme for xed m and a polynomial approximation scheme for arbitrarym. Moreover, we present an approximation algorithm which runs in time O(n logn) for arbitrary m and guarantees a 2i 1 m approximation ratio.

Semi-online scheduling with decreasing job sizes

We investigate the problem of semi-online scheduling jobs on m identical parallel machines where the jobs arrive in order of decreasing sizes. We present a complete solution for the preemptive variant of semi-online scheduling with decreasing job sizes. We give matching lower and upper bounds on the competitive ratio for any fixed number m of machines; these bounds tend to (1+3)/2~1.36603, as the number of machines goes to infinity. Our results are also the best possible for randomized algorithms. For the non-preemptive variant of semi-online scheduling with decreasing job sizes, a result of Graham (SIAM J. Appl. Math. 17 (1969) 263-269) yields a (4/3-1/(3m))-competitive deterministic non-preemptive algorithm. For m=2 machines, we prove that this algorithm is the best possible (it is 7/6-competitive). For m=3 machines we give a lower bound of (1+37)/6~1.18046. Finally, we present a randomized non-preemptive 8/7-competitive algorithm for m=2 machines and prove that this is optimal.

Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines

We give a polynomial approximation scheme for the problem of scheduling on uniformly related parallel machines for a large class of objective functions that depend only on the machine completion times, including minimizing the lp norm of the vector of completion times. This generalizes and simplifies many previous results in this area.

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

We prove a lower bound of the formNΩ(1) on the degree of polynomials in a Nullstellensatz refutation of theCountq polynomials over ℤm, whereq is a prime not dividingm. In addition, we give an explicit construction of a degreeNΩ(1) design for theCountq principle over ℤm. As a corollary, using Beameet al. (1994) we obtain a lower bound of the form 2NΩ(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCountqN from instances of the counting principleCountmM.We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations.Further we shwo that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connectiveMODp follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987) and Smolensky (1987) (in fact, these two proof systems are basically equivalent).

Boolean Circuits, Tensor Ranks, and Communication Complexity

We investigate two methods for proving lower bounds on the size of small-depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that our main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An

First Fit bin packing: A tight analysis.

o(n)

Computer-Aided Complexity Classification of Dial-a-Ride Problems

-bit protocol for a communication game for computing shifts, which also gives an upper bound of

Fast Algorithms for Testing Fault-Tolerance of Sequenced Jobs with Deadlines

o(n^2)

Communication complexity towards lower bounds on circuit depth

on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives an