Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

On the Hardness of Bribery Variants in Voting with CP-Nets

We continue previous work by Mattei et al. (Mattei, N., Pini, M., Rossi, F., Venable, K.: Bribery in voting with CP-nets. Ann. of Math. and Artif. Intell. pp. 1--26 (2013)) in which they study the computational complexity of bribery schemes when voters have conditional preferences that are modeled by CP-nets. For most of the cases they considered, they could show that the bribery problem is solvable in polynomial time. Some cases remained open---we solve two of them and extend the previous results to the case that voters are weighted. Moreover, we consider negative (weighted) bribery in CP-nets, when the briber is not allowed to pay voters to vote for his preferred candidate.Comment: improved readability; identified Cheapest Subsets to be the enumeration variant of K.th Largest Subset, so we renamed it to K-Smallest Subsets and point to the literatur; some more typos fixe

Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks

Understanding the implicit regularization imposed by neural network architectures and gradient based optimization methods is a key challenge in deep learning and AI. In this work we provide sharp results for the implicit regularization imposed by the gradient flow of Diagonal Linear Networks (DLNs) in the over-parameterized regression setting and, potentially surprisingly, link this to the phenomenon of phase transitions in generalized hardness of approximation (GHA). GHA generalizes the phenomenon of hardness of approximation from computer science to, among others, continuous and robust optimization. It is well-known that the $\ell^1$-norm of the gradient flow of DLNs with tiny initialization converges to the objective function of basis pursuit. We improve upon these results by showing that the gradient flow of DLNs with tiny initialization approximates minimizers of the basis pursuit optimization problem (as opposed to just the objective function), and we obtain new and sharp convergence bounds w.r.t.\ the initialization size. Non-sharpness of our results would imply that the GHA phenomenon would not occur for the basis pursuit optimization problem -- which is a contradiction -- thus implying sharpness. Moreover, we characterize $\textit{which}$ $\ell_1$ minimizer of the basis pursuit problem is chosen by the gradient flow whenever the minimizer is not unique. Interestingly, this depends on the depth of the DLN

The complexity of general-valued CSPs seen from the other side

The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small correction

Circular economy implementation in waste management network design problem: a case study

The paper presents a new approach to support the strategic decision-making in the area of municipal solid waste management applying modern circular economy principles. A robust two-stage integer non-linear program is developed. The primary goal tends to reduce the waste production. The generated waste should be preferably recycled as much as possible and the resultant residual waste might be used for energy recovery. Only some waste residues are appropriate for landfilling. The aim is to propose the near-optimal waste allocation for its suitable processing as well as waste transportation plan at an operational level. In addition, the key strategical decisions on waste treatment facilities location must be made. Since waste production is very often hard to predict, it is modeled as an uncertain decision-dependent quantity. To support the circular economy ideas, advertising and pricing principles are introduced and applied. Due to the size of available real-world data and complexity of the designed program, the presented model is linearized and uncertainty is handled by a robust optimization methodology. The model, data, and algorithm are implemented in MATLAB and Julia, using the state-of-the-art solvers. The computational result is a set of decisions providing a trade-off between the average performance and the immunization against the worst-case conditions. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature