Seminar on Approximation and Online Algorithms

**5. 1. 2018** (pátek [Friday], 10:40, MFF UK Malá Strana S6)

**Moses Charikar, Konstantin Makarychev, Yury Makarychev: Integrality
Gaps
for Sherali-Adams Relaxations.** STOC 2009.

(presented by Jakub Sosnovec)

**Abstract: **We prove strong lower bounds on integrality
gaps of Sherali-Adams relaxations for MAX CUT, Vertex Cover,
Sparsest Cut and other problems. Our constructions show gaps for
Sherali-Adams relaxations that survive n^{δ} rounds of
lift and project. For MAX CUT and Vertex Cover, these show that
even n^{δ} rounds of Sherali-Adams do not yield a better
than 2-ε approximation. The main combinatorial challenge in
constructing these gap examples is the construction of a
fractional solution that is far from an integer solution, but yet
admits consistent distributions of local solutions for all small
subsets of variables. Satisfying this consistency requirement is
one of the major hurdles to constructing Sherali-Adams gap
examples. We present a modular *recipe* for achieving this,
building on previous work on metrics with a local-global
structure. We develop a conceptually simple geometric approach to
constructing Sherali-Adams gap examples via constructions of
consistent *local* SDP solutions. This geometric approach is
surprisingly versatile. We construct Sherali-Adams gap examples
for Unique Games based on our construction for MAX CUT together
with a parallel repetition like procedure. This in turn allows us
to obtain Sherali-Adams gap examples for any problem that has a
Unique Games based hardness result (with some additional
conditions on the reduction from Unique Games). Using this, we
construct 2-ε gap examples for Maximum Acyclic Subgraph that rules
out any family of linear constraints with support at most n^{δ}.

**Ola Svensson, Jakub Tarnawski, László A. Végh: A Constant-Factor
Approximation Algorithm for the Asymmetric Traveling Salesman
Problem.** https://arxiv.org/abs/1708.04215.

(presented by Pavel Veselý)

**Abstract: **We give a constant-factor approximation
algorithm for the asymmetric traveling salesman problem. Our
approximation guarantee is analyzed with respect to the standard
LP relaxation, and thus our result confirms the conjectured
constant integrality gap of that relaxation.

Our techniques build upon the constant-factor approximation
algorithm for the special case of node-weighted metrics.
Specifically, we give a generic reduction to structured instances
that resemble but are more general than those arising from
node-weighted metrics. For those instances, we then solve
Local-Connectivity ATSP, a problem known to be equivalent (in
terms of constant-factor approximation) to the asymmetric
traveling salesman problem.

**8. 12. ****2017** (pátek [Friday], 10:40, MFF UK Malá
Strana S6)

**Andreas Emil Feldmann: Parameterized approximation algorithms
for directed Steiner network problems in bidirected graphs
**

For problems related to DSN it has been fruitful before to analyze the structure of the optimum in order to obtain approximation and FPT algorithms. Therefore, we consider the DSN_K problem, where we need to compute the optimum network contained in some class K of graphs. If K for instance is the class of planar graphs, then this generalizes several well-studied special cases, such as DSN on planar input graphs, but also the Directed Steiner Tree problem on general input graphs. Our main result proves that if additionally the input is bidirected, then a parameterized approximation scheme exists if we aim to compute the planar optimum. We also provide several hardness results that show that our obtained runtime is the best possible for this special case, while at the same time no parameterized approximation scheme exists for any generalization of this special case. We also consider the Strongly Connected Steiner Subgraph (SCSS) problem as another special case, which cannot be captured by DSN_K for any non-trivial class K. We prove that a known parameterized 2-approximation is best possible for SCSS, and that on bidirected input graphs the problem is FPT for parameter k.

**10. 11., 24. 11. 2017** (pátek [Friday], 10:40, MFF UK Malá
Strana S6)

**Ola Svensson: Approximating
ATSP by Relaxing Connectivity. FOCS 2015. **Also https://arxiv.org/abs/1502.02051.

(presented by Martin Böhm)

**Abstract: **The standard LP relaxation of the asymmetric
traveling salesman problem has been conjectured to have a constant
integrality gap in the metric case. We prove this conjecture when
restricted to shortest path metrics of node-weighted digraphs. Our
arguments are constructive and give a constant factor
approximation algorithm for these metrics. We remark that the
considered case is more general than the directed analog of the
special case of the symmetric traveling salesman problem for
which there were recent improvements on Christofides' algorithm.

**27. 10., 3. 11. 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.