# Seminář z aproximačních a online algoritmů Seminar on Approximation and Online Algorithms

## Petr Kolman, Jiří Sgall

### Předchozí program semináře, zimní semestr 2017 [Previous program, Fall 2017]

3. 11., 27. 10. 2017 (pátek [Friday], 10:40, MFF UK Malá Strana S6)

Syamantak Das and Andreas Wiese: On minimizing the makespan when some jobs cannot be assigned on the same machine. ESA 2017.
(presented by Jan Voborník)

Abstract: We study the classical scheduling problem of assigning jobs to machines in order to minimize the makespan. It is well-studied and admits an EPTAS on identical machines and a (2-1/m)-approximation algorithm on unrelated machines. In this paper we study a variation in which the input jobs are partitioned into bags and no two jobs from the same bag are allowed to be assigned on the same machine. Such a constraint can easily arise, e.g., due to system stability and redundancy considerations. Unfortunately, as we demonstrate in this paper, the techniques of the above results break down in the presence of these additional constraints. Our first result is a PTAS for the case of identical machines. It enhances the methods from the known (E)PTASs by a finer classification of the input jobs and careful argumentations why a good schedule exists after enumerating over the large jobs. For unrelated machines, we prove that there can be no (log n)^{1/4-epsilon}-approximation algorithm for the problem for any epsilon > 0, assuming that NP nsubseteq ZPTIME(2^{(log n)^{O(1)}}). This holds even in the restricted assignment setting. However, we identify a special case of the latter in which we can do better: if the same set of machines we give an 8-approximation algorithm. It is based on rounding the LP-relaxation of the problem in phases and adjusting the residual fractional solution after each phase to order to respect the bag constraints.

13. 10., 20. 10. 2017 (pátek [Friday], 10:40, MFF UK Malá Strana S6)

Pavel Veselý: Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
(joint work with Pavel Dvořák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masařík, and Tomáš Toufar)

Abstract: We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This probl em has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX- hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the assumed p arameter.

We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPTAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]- hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

6. 10., 13. 10. 2017 (pátek [Friday], 10:40, MFF UK Malá Strana S6)

Petr Kolman: The length bounded cut problem

Abstract: Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer parameter $L$, an $L$-bounded cut is a subset $F$ of edges such that the every path between $s$ and $t$ in $(V,E-F)$ has length at least $L$. The minimum $L$-bounded cut problem is to find an $L$-bounded cut of minimum cardinality; the problem is known also under the name the short paths interdiction problem.

The early research on the problem was motivated by an effort to generalize the Max Flow - Min Cut theorem to flows of bounded lenght (i.e., flows decomposable into paths of bounded length). Another motivation for studying the problem were military applications: the goal was to disrupt the movements of enemy troops and material in a capacitated supply network. Later, the problem was also studied in the context of communication networks where the length of paths is often an issue.

Though the problem is very simple to state and has been studied since the beginning of the 70's, it is not much understood yet. It is known that the problem is NP-hard and that even finding a 1.1377 approximation is NP-hard. On the other hand, the best known approximation algorithm for general graphs has approximation ratio, in terms of $n$, only $\Theta(n^{2/3})$.

We will survey known results about the problem and present several new ones. In particular, we will show that for planar and for bounded genus graphs, the problem is fixed parameter tractable (FPT) with respect to $L$, and for graphs of treewidth bounded by $\tau$, we describe $\tau$-approximation algorithm for a vertex version of the problem. We will also discuss related open problems.