Seminář z aproximačních a online algoritmů
Petr Kolman, Jiří Sgall
Předchozí program semináře, letní semestr 2011 [Previous
program, Spring 2011]
17. 5., 24. 5. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Dániel Marx and Igor Razgon: Fixed-parameter
tractability of multicut parameterized by the size of the cutset. STOC
2011.
(referuje Dušan Knop)
Abstract: Given an undirected graph $G$, a collection
$\{(s_1,t_1),..., (s_k,t_k)\}$ of
pairs of vertices, and an integer $p$, the Edge Multicut problem ask if
there
is a set $S$ of at most $p$ edges such that the removal of $S$
disconnects
every $s_i$ from the corresponding $t_i$. Vertex Multicut is the
analogous
problem where $S$ is a set of at most $p$ vertices. Our main result is
that
both problems can be solved in time $2^{O(p^3)}... n^{O(1)}$, i.e.,
fixed-parameter tractable parameterized by the size $p$ of the cutset
in the
solution. By contrast, it is unlikely that an algorithm with running
time of
the form $f(p)... n^{O(1)}$ exists for the directed version of the
problem, as
we show it to be W[1]-hard parameterized by the size of the cutset.
3. 5., 10. 5. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Friedrich Eisenbrand, Dömötör Pálvölgyi, and Thomas
Rothvoss:
Bin Packing via Discrepancy of Permutations. SODA
2011.
(referuje Tomáš Dzetkulič)
Abstract: A well studied special case of bin packing is the
3-partition problem, where n items of size >1/4 have to be packed in
a minimum number of bins of capacity one. The famous Karmarkar-Karp
algorithm transforms a fractional solution of a suitable LP relaxation
for this problem into an integral solution that requires at most O(log
n) additional bins.
The three-permutations-conjecture of Beck is the following. Given any 3
permutations on n symbols, one can color the symbols red and blue, such
that in any interval of any of those permutations, the number of red
and blue symbols differs only by a constant. Beck's conjecture is well
known in the field of discrepancy theory.
We establish a surprising connection between bin packing and Beck's
conjecture: If the latter holds true, then the additive integrality gap
of the 3-partition linear programming relaxation is bounded by a
constant.
19. 4., 26. 4. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Ola Svensson: Santa
Claus Schedules Jobs on Unrelated Machines. STOC 2010.
(referuje Marek Krčál)
Abstract: One of the classic
results in scheduling theory is the 2-approximation
algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling
jobs to
minimize makespan on unrelated machines, i.e., job j requires time
p_{ij} if
processed on machine i. More than two decades after its introduction it
is
still the algorithm of choice even in the restricted model where
processing
times are of the form p_{ij} in {p_j, \infty}. This problem, also known
as the
restricted assignment problem, is NP-hard to approximate within a
factor less
than 1.5 which is also the best known lower bound for the general
version. Our main result is a polynomial time algorithm that estimates
the optimal
makespan of the restricted assignment problem within a factor 33/17 +
\epsilon
\approx 1.9412 + \epsilon, where \epsilon > 0 is an arbitrary small
constant.
The result is obtained by upper bounding the integrality gap of a
certain
strong linear program, known as configuration LP, that was previously
successfully used for the related Santa Claus problem. Similar to the
strongest
analysis for that problem our proof is based on a local search
algorithm that
will eventually find a schedule of the mentioned approximation
guarantee, but
is not known to converge in polynomial time.
12. 4. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Robin A. Moser and Dominik Scheder: A Full Derandomization of
Schoening's k-SAT Algorithm. STOC 2011.
(referuje Vojtěch Bardejovský)
Abstract: Schoening [7] presents a simple randomized algorithm
for k-SAT with running time O(a_k^n poly(n)) for a_k = 2(k − 1)/k. We
give a deterministic version of this algorithm running in time O((a_k +
eps)n poly(n)), where eps>0 can be made arbitrarily small.
29. 3., 5. 4. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Georg Gottlob,
Reinhard Pichler,
Fang Wei:
Monadic datalog over
finite structures of bounded treewidth.
ACM Trans. Comput. Log. 12(1): 3 (2010)
(referuje Martin Koutecký)
Abstract: Namísto referování nových výsledků v článku se
zaměříme především na v něm obsažený nový důkaz Courcellovy věty, která
říká že formule v MSOL (monadické logice druhého řádu) je rozhodnutelná
nad grafem (strukturou) s omezenou stromovou šířkou v lineárním čase.
15. 3., 22.3. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Kevin P. Costello, Asaf Shapira, and Prasad Tetali: Randomized greedy: new variants of some classic
approximation algorithms. SODA 2011.
(referuje Ondřej Zajíček)
Abstract: We consider the performance of two classic
approximation algorithms which work by scanning the input and greedily
constructing a solution. We investigate whether running these
algorithms on a random permutation of the input can increase their
performance ratio. We obtain the following results:
1. Johnson’s approximation algorithm for MAX-SAT is one of the first
approximation algorithms to be rigorously analyzed. It has been shown
that the performance ratio of this algorithm is 2/3. We show that when
executed on a random permutation of the variables, the performance
ratio of this algorithm is improved to 2/3 + c for some c > 0 This
resolves an open problem of Chen, Friesen and Zhang [JCSS 1999]. (See
also the paper by Poloczek and Schnitger in these proceedings for
related results on this algorithm and its variants).
2. Motivated by the above improvement, we consider the performance of
the greedy algorithm for MAXCUT whose performance ratio is 1/2. Our
hope was that running the greedy algorithm on a random permutation of
the vertices would result in a 1/2 + c approximation algorithm.
However, it turns out that in this case the performance of the
algorithm remains 1/2. This resolves an open problem of Mathieu and
Schudy [SODA 2008].
1. 3., 8. 3. 2011 (úterý [Tuesday], 8:30,
MFF UK Malá
Strana, S6)
Davide Bilo, Luciano Guala and Guido Proietti: Improved
approximability
and
non-approximability
results
for
graph
diameter
decreasing
problems. MFCS 2010.
(referuje Petr Kolman)
Abstract: In this paper we study two variants of the problem
of adding edges to a graph so as to reduce the resulting diameter. More
precisely, given a graph G = (V,E), and two positive integers D and B,
the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem
is to find a minimum cardinality set F of edges to be added to G in
such a way that the diameter of G+F is less than or equal to D, while
the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem
is to find a set F of B edges to be added to G in such a way that the
diameter of G+F is minimized. Both problems are well known to be
NP-hard, as well as approximable within O(log n log D) and 4 (up to an
additive term of 2), respectively. In this paper, we improve these
longstanding approximation ratios to O(log n) and to 2 (up to an
additive term of 2), respectively. As a consequence, we close, in an
asymptotic sense, the gap on the approximability of the MCBD problem,
which was known to be not approximable within c log n, for some
constant c > 0, unless P = NP. Remarkably, as we further show in the
paper, our approximation ratio remains asymptotically tight even if we
allow for a solution whose diameter is optimal up to a multiplicative
factor approaching 5/3. On the other hand, on the positive side, we
show that at most twice of the minimal number of additional edges
suffices to get at most twice of the required diameter.