Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. In this paper we continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where pmax is the maximum processing time of a job and wmax is the maximum weight of a job:-Makespan minimization on uniformly related machines (Q||Cmax) parameterized by pmax, -Makespan minimization on unrelated machines (R||Cmax) parameterized by pmax and the number of kinds of machines (defined later), -Sum of weighted completion times minimization on unrelated machines (R|| wjCj) parameterized by pmax + wmax and the number of kinds of machines, -The same problem, R|| wjCj, parameterized by the number of distinct job times and the number of machines.

Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixedparameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs.Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a singleexponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both.Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a "local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.1 Kannan's algorithm in parameterized complexity: they modeled Closest String with k input strings as an ILP of dimension k O(k) , and thereby concluded with the first fixed-parameter algorithm for Closest String. This success led Niedermeier [Nie04] to propose in his book:[...] It remains to investigate further examples besides Closest String where the described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer) linear programming.Since then, many more applications of Lenstra's and Kannan's algorithm for parameterized problems have been proposed. However, essentially all of them [BFN + 15, DS12, FLM + 08, HR15, Lam12, MW15] share a common trait with the algorithm for Closest String: they have a doubly-exponential run time dependence on the parameter. Moreover, it is difficult to find ILP formulations with small dimension for problems whose input contains many objects with varying cost functions, such as in Swap Bribery [BCF + 14, Challenge #2].

We introduce a general problem about bribery in voting systems. In the R-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the manipulated election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count for a candidate. As our main result, we show that R-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules R, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized of R-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the 9 Challenges in computational social choice by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the unit-cost case for all scoring protocols, the maximin rule, and Bucklin rule.

We study the effects of campaigning, where the society is partitioned into voter clusters and a diffusion process propagates opinions in a network connecting those clusters. Our model is very general and can incorporate many campaigning actions, various partitions of the society into voter clusters, and very general diffusion processes. Perhaps surprisingly, we show that computing the cheapest campaign for rigging a given election can usually be done efficiently, even with arbitrarily-many voters.