On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs

  title={On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs},
  author={Kim-Manuel Klein and Adam Polak and Lars Rohwedder},
The starting point of this paper is the problem of scheduling n jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., 1 | | P p j U j . This problem was identified by Bringmann et al. (Algo-rithmica 2022) as a natural subquadratic-time special case of the classic 1 | | P w j U j problem, which likely requires time quadratic in the total processing time P , because of a fine-grained lower bound. Bringmann et al. obtain their e… 

Figures from this paper

Quick Minimization of Tardy Processing Time on a Single Machine

The running time of the resulting algorithm is equivalent, up to logarithmic factors, to the time it takes to compute a ( max, min ) -skewed-convolution of two vectors of integers whose sum is O ( P ) , where P is the sum of the jobs’ processing times.



Single Machine Weighted Number of Tardy Jobs Minimization With Small Weights

This paper presents an algorithm for the classical scheduling problem of minimizing the weighted number of jobs on a single machine, the so-called 1 || ∑ wjUj problem, and proves new results concerning pseudo-polynomial time algorithms assuming ∀∃-SETH conjecture, a recently introduced variant of the well known Strong Exponential Time Hypothesis (SETH).

Scheduling meets n-fold integer programming

This paper shows that several additional cases of fundamental scheduling problems are fixed-parameter tractable for some natural parameters, including n-fold integer programming, a recent variable dimension technique which the author believes to be highly relevant for the parameterized complexity community.

Knapsack and Subset Sum with Small Items

These algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance.

Faster Minimization of Tardy Processing Time on a Single Machine

Two new algorithms are developed for the problem of minimizing the total processing time of tardy jobs on a single machine, each improving on Lawler and Moore’s algorithm in a different scenario.

Efficient sequential and parallel algorithms for multistage stochastic integer programming using proximity

It is proved that if the authors consider an integer program P, say with a constraint matrix A, then for every optimum solution to the linear relaxation of P there exists an optimum (integral) solution to P that lies, in the 𝓁_{∞}-norm, within distance bounded by a function of ‖A‖‖_∞ and the primal treedepth of A.

Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max

It is proved that approximating directed APSP and exactly computing the Min-Max Product are equivalent, and the first strongly polynomial approximation scheme for Min-Plus Convolution in strongly subquadratic time is obtained and an equivalence of approximate Min- plus Convolution and exact Min- Max Convolution is proved.

On the Fine-grained Complexity of One-Dimensional Dynamic Programming

Subquadratic equivalences are proved between the following pairs (an LWS instantiation and the corresponding core problem) of problems: a low-rank version of LWS and minimum inner product, finding the longest chain of nested boxes and vector domination, and a coin change problem which is closely related to the knapsack problem and (min,+)-convolution.

On Problems Equivalent to (min,+)-Convolution

This article establishes the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences and explains why replacing this assumption with the Strong Exponential Time Hypothesis might not be possible for some problems.

A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum

A simple and elegant randomized algorithm running in time O(n+t) that improves upon a classic algorithm and is likely to be near-optimal, since it matches conditional lower bounds from S et C over and k-C lique .

Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

This paper shows for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem, and gives the first fine-grained equivalence between natural problems of different running times.