Jiří Sgall

Solution of David Gale's lion and man problem

A pursue-and-evasion game is analyzed, including almost optimal bounds on the number of moves needed to win.

Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Abstract. Razborov (1996) recently proved that polynomial calculus proofs of the pigeonhole principle

A lower bound for randomized on-line multiprocessor scheduling

P H P^m_n

Graph balancing: a special case of scheduling unrelated parallel machines

must have degree at least ⌈n/2⌉ + 1 over any field. We present a simplified proof of the same result.¶Furthermore, we show a matching upper bound on polynomial calculus proofs of the pigeonhole principle for any field of suficiently large characteristic, and an ⌈n/2⌉ + 1 lower bound for any subset sum problem over the field of reals.¶We show that these degree lower bounds also translate into lower bounds on the number of monomials in any polynomial calculus proof, and hence on the running time of most implementations of the Gröbner basis algorithm.

Ancient and new algorithms for load balancing in the L p norm

We significantly improve the previous lower bounds on the performance of randomized algorithms for on-line scheduling jobs on m identical machines. We also show that a natural idea for constructing an algorithm with matching performance does not work.

Improved online algorithms for buffer management in QoS switches

We design a 1.75-approximation algorithm for a special case of scheduling parallel machines to minimize the makespan, namely the case where each job can be assigned to at most two machines, with the same processing time on either machine. (This is a special case of so-called restricted assignment, where the set of eligible machines can be arbitrary for each job.) This is the first improvement of the approximation ratio 2 of Lenstra, Shmoys, and Tardos (Math. Program. 46:259–271, 1990), to a smaller constant for any special case with unbounded number of machines and unbounded processing times. We conclude by showing integrality gaps of several relaxations of related problems.

The greedy algorithm for the minimum common string partition problem

Abstract. We consider the on-line load balancing problem where there are m identical machines (servers) and a sequence of jobs. The jobs arrive one by one and should be assigned to one of the machines in an on-line fashion. The goal is to minimize the sum (over all machines) of the squares of the loads, instead of the traditional maximum load. We show that for the sum of the squares the greedy algorithm performs within 4/3 of the optimum, and no on-line algorithm achieves a better competitive ratio. Interestingly, we show that the performance of Greedy is not monotone in the number of machines. More specifically, the competitive ratio is 4/3 for any number of machines divisible by 3 but strictly less than 4/3 in all the other cases (although it approaches 4/3 for a large number of machines). To prove that Greedy is optimal, we show a lower bound of 4/3 for any algorithm for three machines. Surprisingly, we provide a new on-line algorithm that performs within 4/3 -δ of the optimum, for some fixed δ>0 , for any sufficiently large number of machines. This implies that the asymptotic competitive ratio of our new algorithm is strictly better than the competitive ratio of any possible on-line algorithm. Such phenomena is not known to occur for the classic maximum load problem. Minimizing the sum of the squares is equivalent to minimizing the load vector with respect to the l2 norm. We extend our techniques and analyze the exact competitive ratio of Greedy with respect to the lp norm. This ratio turns out to be 2 - Θ(( ln p)/p) . We show that Greedy is optimal for two machines but design an algorithm whose asymptotic competitive ratio is better than the ratio of Greedy.

A lower bound for on-line scheduling on uniformly related machines

We consider the following buffer management problem arising in QoS networks: Packets with specified weights and deadlines arrive at a network switch and need to be forwarded so that the total weight of forwarded packets is maximized. Packets not forwarded before their deadlines are lost. The main result of the article is an online 64/33 ≈ 1.939-competitive algorithm, the first deterministic algorithm for this problem with competitive ratio below 2. For the 2-uniform case we give an algorithm with ratio ≈ 1.377 and a matching lower bound.

Online Competitive Algorithms for Maximizing Weighted Throughput of Unit Jobs

In the Minimum Common String Partition problem (MCSP), we are given two strings on input, and we wish to partition them into the same collection of substrings, minimizing the number of the substrings in the partition. This problem is NP-hard, even for a special case, denoted 2-MCSP, where each letter occurs at most twice in each input string. We study a greedy algorithm for MCSP that at each step extracts a longest common substring from the given strings. We show that the approximation ratio of this algorithm is between Ω(n0.43) and O(n0.69). In the case of 2-MCSP, we show that the approximation ratio is equal to 3. For 4-MCSP, we give a lower bound of Ω(log n).

Preemptive online scheduling: optimal algorithms for all speeds

We consider the problem of on-line scheduling of jobs arriving one by one on uniformly related machines, with or without preemption. We prove a lower bound of 2, both with and without preemption, for randomized algorithms working for an arbitrary number of machines. For a constant number of machines we give new lower bounds for the preemptive case.