**18. 5. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá Strana
S8)

**Marcin Bienkowski: An
Optimal
Lower Bound for Buffer Management in Multi-Queue Switches.**
Algorithmica 68:426-447, 2012. Preliminary version SODA
2011.

**Abstract:**** **In the online packet buffering problem
(also known as the unweighted FIFO variant of
buffer management), we focus on a single network packet switching
device with several input ports and one output port. This device
forwards unit-size, unit-value packets from input ports to the
output port. Buffers attached to input ports may accumulate
incoming packets for later transmission; if they cannot
accommodate all incoming packets, their excess is lost. A packet
buffering algorithm has to choose from which buffers to transmit
packets in order to minimize the number of lost packets and thus
maximize the throughput.

We present a tight lower bound of e/(e−1)≈1.582 on the competitive ratio of the throughput maximization, which holds even for fractional or randomized algorithms. This improves the previously best known lower bound of 1.4659 and matches the performance of the algorithm Random Schedule. Our result contradicts the claimed performance of the algorithm Random Permutation; we point out a flaw in its original analysis.

**11. 5. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá Strana
S8)

**Jan Elffers, Mathijs de Weerdt: ****Scheduling with two
non-unit task lengths is NP-complete.**

**27. 4., 4. 5. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá
Strana S8)

**Morteza Monemizadeh: ****Prophet Secretary**

(joint work with Hossein Esfandiari, MohammadTaghi Hajiaghayi, and
Vahid Liaghat)

**Abstract:** Optimal stopping theory is a powerful tool for
analyzing scenarios such as online auctions in which we generally
require optimizing an objective function over the space of
stopping rules for an allocation process under uncertainty.
Perhaps the most classic problems of stopping theory are the
prophet inequality problem and the secretary problem. The
classical prophet inequality states that by choosing the same
threshold OPT/2 for every step, one can achieve the tight
competitive ratio of 0.5. On the other hand, for the basic
secretary problem, the optimal strategy is to ignore the first n/e
elements of the sequence while using the maximum of the first n/e
elements as the threshold for the rest of sequence. This strategy
achieves the optimal competitive ratio of 1/e=0.36.

In this paper, we introduce prophet secretary, a natural combination of the prophet inequality problem and the secretary problem. An example of motivations for our problem is as follows. Consider a seller that has an item to sell on the market to a set of arriving customers. The seller knows the types of customers that may be interested in the item and he has a price distribution for each type: the price offered by a customer of a type is anticipated to be drawn from the corresponding distribution. However, the customers arrive in a random order. Upon the arrival of a customer, the seller makes an irrevocable decision to whether sell the item at the offered price. We address thequestion of finding a strategy for selling the item at a high price.

We show that by using a single uniform threshold one cannot break the 0.5 barrier of the prophet inequality for the prophet secretary problem. However, we show that

- using n distinct non-adaptive thresholds one can indeed obtain the competitive ratio of (1-1/e \approx 0.63); and
- no online algorithm can achieve a competitive ratio better
than 0.73.

Our results improve the approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1-1/e) when the order of agents (customers) are chosen randomly.

We also consider the minimization variants of the prophet inequality problem. In particular, we show that, even for the simple case in which numbers are drawn from identical and independent distributions (i.i.d.), there is no constant competitive online algorithm for the minimization variants of the prophet inequality and prophet secretary problems.**13. 4. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá Strana
S8)

**Andreas Björklund and Thore Husfeldt: Shortest
Two Disjoint Paths in Polynomial Time. **ICALP 2014
(best paper award).

(referuje Jan Voborník)

**Abstract**: Given an undirected graph and two pairs of
vertices (s_i ,t_i) for i ∈ {1,2} we show that there is a
polynomial time Monte Carlo algorithm that finds disjoint paths of
smallest total length joining s_i and t_i for i ∈ {1,2}
respectively, or concludes that there most likely are no such
paths at all. Our algorithm applies to both the vertex- and
edge-disjoint versions of the problem. Our algorithm is algebraic
and uses permanents over the quotient ring Z_4[X]/(X^m) in
combination with Mulmuley, Vazirani and Vazirani’s isolation lemma
to detect a solution. We develop a fast algorithm for permanents
over said ring by modifying Valiant’s 1979 algorithm for the
permanent over Z_{2^t}.

**30. 3., 20. 4. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá
Strana S8)

**Nikhil Bansal: Approximating
independent sets in sparse graphs. **SODA 2015.

(referuje Martin Böhm)

**Abstract:** We consider the maximum independent set problem
on sparse graphs with maximum degree d. The best known result for
the problem is an SDP based O(d log log d / log d) approximation
due to Halperin. It is also known that no o(d / log^2 d)
approximation exists assuming the Unique Games Conjecture. We show
the following two results:

(i) The natural LP formulation for the problem strengthened by
O(log^4 d) levels of the mixed-hierarchy has an integrality gap of
O~(d / log^2 d), where O~() ignores some log log d factors.
However, our proof is non-constructive, in particular it uses an
entropy based approach due to Shearer, and does not give a O~(d /
log2 d) approximation algorithm with sub-exponential running time.

(ii) We give an O(d / log d) approximation based on
polylog(d)-levels of the mixed hierarchy that runs in n^O(1)
exp(log^O(1) d) time, improving upon Halperin's bound by a modest
log log d factor. Our algorithm is based on combining Halperin's
approach together with an idea used by Ajtai, Erdos, Komlos and
Szemeredi to show that K_r-free, degree-d graphs have independent
sets of size Omega_r(n log log d / d).

**23. 3. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá Strana
S8)

**Yann Disser and Martin Skutella: The Simplex Algorithm is
NP-mighty. SODA 2015. Also http://arxiv.org/abs/1311.5935.**

(referuje Pavel Veselý)

**Abstract:** We propose to classify the power of algorithms
by the complexity of the problems that they can be used to solve.
Instead of restricting to the problem a particular algorithm was
designed to solve explicitly, however, we include problems that,
with polynomial overhead, can be solved 'implicitly' during the
algorithm's execution. For example, we allow to solve a decision
problem by suitably transforming the input, executing the
algorithm, and observing whether a specific bit in its internal
configuration ever switches during the execution. We show that the
Simplex Method, the Network Simplex Method (both with Dantzig's
original pivot rule), and the Successive Shortest Path Algorithm
are NP-mighty, that is, each of these algorithms can be used to
solve any problem in NP. This result casts a more favorable light
on these algorithms' exponential worst-case running times.
Furthermore, as a consequence of our approach, we obtain several
novel hardness results. For example, for a given input to the
Simplex Algorithm, deciding whether a given variable ever enters
the basis during the algorithm's execution and determining the
number of iterations needed are both NP-hard problems. Finally, we
close a long-standing open problem in the area of network flows
over time by showing that earliest arrival flows are NP-hard to
obtain.

**9. 3., 16. 3. 2015 **(Pondělí [Monday], 10:40, MFF UK Malá
Strana S8)

**Gábor Braun, Sebastian Pokutta: The matching polytope does not
admit fully-polynomial size relaxation schemes. SODA 2015. Also
http://arxiv.org/abs/1403.6710.**

(referuje Martin Koutecký)

**Abstract:** The groundbreaking work of Rothvoss [arxiv:1311.2369] established
that every linear program expressing the matching polytope has an
exponential number of inequalities (formally, the matching
polytope has exponential extension complexity). We generalize this
result by deriving strong bounds on the polyhedral
inapproximability of the matching polytope: for fixed 0<ε<1,
every polyhedral (1+ε/n)-approximation requires an exponential
number of inequalities, where n is the number of vertices. This is
sharp given the well-known ρ-approximation of size O((nρ/(ρ−1)))
provided by the odd-sets of size up to ρ/(ρ−1). Thus matching is
the first problem in P, whose natural linear encoding does not
admit a fully polynomial-size relaxation scheme (the polyhedral
equivalent of an FPTAS), which provides a sharp separation from
the polynomial-size relaxation scheme obtained e.g., via
constant-sized odd-sets mentioned above.

Our approach reuses ideas from Rothvoss [arxiv:1311.2369], however
the main lower bounding technique is different. While the original
proof is based on the hyperplane separation bound (also called the
rectangle corruption bound), we employ the information-theoretic
notion of common information as introduced in Braun and Pokutta [this http URL],
which
allows to analyze perturbations of slack matrices. It turns out
that the high extension complexity for the matching polytope stem
from the same source of hardness as for the correlation polytope:
a direct sum structure.

**23. 2. 2015, 2. 3. 2015 **(Pondělí [Monday], 10:40, MFF UK
Malá Strana S8)

**Lukáš Folwarczný: Caching Is Hard: Even For Small Pages**

**Abstract:** General caching (pages have arbitrary sizes and
fault costs) in the offline setting has been studied since 1990s.
In 2010, it was shown that general caching is strongly NP-hard,
even when all pages have fault cost one (this case is known as the
fault model). However, pages of arbitrary size (e.g. as big as
half of the cache) are needed to prove that the problem is hard.
In real-life problems, pages are likely to be small in comparison
with the size of the cache. We present a new reduction that proves
the strong NP-hardness of caching in the fault model with page
sizes restricted to {1, 2, 3}.