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In the online balanced graph repartitioning problem, one has to maintain a clustering of $n$ nodes into $\ell$ clusters, each having $k = n / \ell$ nodes. During runtime, an online algorithm is given a stream of communication requests between pairs of nodes: an inter-cluster communication costs one unit, while the intra-cluster communication is free. An algorithm can change the clustering, paying unit cost for each moved node. This natural problem admits a simple $O(\ell^2 \cdot k^2)$-competitive algorithm COMP, whose performance is far apart from the best known lower bound of $\Omega(\ell \cdot k)$. One of open questions is whether the dependency on $\ell$ can be made linear; this question is of practical importance as in the typical datacenter application where virtual machines are clustered on physical servers, $\ell$ is of several orders of magnitude larger than $k$. We answer this question affirmatively, proving that a simple modification of COMP is $(\ell \cdot 2^{O(k)})$-competitive. On the technical level, we achieve our bound by translating the problem to a system of linear integer equations and using Graver bases to show the existence of a ``small'' solution.

The theory of $n$-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an $n$-fold integer program (IP) consists of parameter $A$, dimension $n$, and numerical data of binary encoding length $L$. It was known for some time that such programs can be solved in polynomial time using $O(n^{g(A)}L)$ arithmetic operations where $g$ is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable (FPT) time using $O(f(A)n^3L)$ arithmetic operations for a single-exponential function $f$. This, and a faster algorithm for a special case of combinatorial $n$-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using $O(n^{g(A)})$ arithmetic operations. Here we establish a result which subsumes all three of the above results by showing that $n$-fold IP can be solved in strongly polynomial FPT time using $O(f(A)n^3)$ arithmetic operations. In fact, our results are much more general, briefly outlined as follows. - There is a strongly polynomial algorithm for ILP whenever a so-called Graver-best oracle is realizable for it. - Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in FPT time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial. - We show that ILP is FPT parameterized by the largest coefficient $\|A\|_\infty$ and the primal or dual treedepth of $A$, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.

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 fixed-parameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problemby 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 single-exponential 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., 245 pages, preliminary results were presented during ESA 2017