Dzmitry Sledneu

Lawler's minmax cost algorithm: optimality conditions and uncertainty

The well-known

A combinatorial algorithm for all-pairs shortest paths in directed vertex-weighted graphs with applications to disc graphs

O(n^2)

A QPTAS for the Base of the Number of Crossing-Free Structures on a Planar Point Set

O(n2) minmax cost algorithm of Lawler (MANAGE SCI 19(5):544–546, 1973) was developed to minimize the maximum cost of jobs processed by a single machine under precedence constraints. We propose two results related to Lawler’s algorithm. Lawler’s algorithm delivers one specific optimal schedule while there can exist other optimal schedules. We present necessary and sufficient conditions for the optimality of a schedule for the problem with strictly increasing cost functions. The second result concerns the same scheduling problem under uncertainty. The cost function for each job is of a special decomposable form and depends on the job completion time and on an additional numerical parameter, for which only an interval of possible values is known. For this problem we derive an

Bounds for Semi-disjoint Bilinear Forms in a Unit-Cost Computational Model

O(n^2)

Detecting monomials with k distinct variables

O(n2) algorithm for constructing a schedule that minimizes the maximum regret criterion . To obtain this schedule, we use Lawler’s algorithm as a part of our technique.

Optimal Cuts and Bisections on the Real Line in Polynomial Time

A straightforward natural iterative heuristic for correlation clustering in the general setting is to start from singleton clusters and whenever merging two clusters improves the current quality score merge them into a single cluster. We analyse the approximation and complexity aspects of this heuristic and its three simple deterministic or random refinements.