49 research outputs found
Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
The most well-known conjecture in the context of matroid secretary problems
claims the existence of a constant-factor approximation applicable to any
matroid. Whereas this conjecture remains open, modified forms of it were shown
to be true, when assuming that the assignment of weights to the secretaries is
not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and
Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid
secretary problem with adversarial weight assignment for which a
constant-factor approximation was found. We address this point by presenting a
9-approximation for the \emph{free order model}, a model suggested shortly
after the introduction of the matroid secretary problem, and for which no
constant-factor approximation was known so far. The free order model is a
relaxed version of the original matroid secretary problem, with the only
difference that one can choose the order in which secretaries are interviewed.
Furthermore, we consider the classical matroid secretary problem for the
special case of laminar matroids. Only recently, a constant-factor
approximation has been found for this case, using a clever but rather involved
method and analysis (Im and Wang, [SODA 2011]) that leads to a
16000/3-approximation. This is arguably the most involved special case of the
matroid secretary problem for which a constant-factor approximation is known.
We present a considerably simpler and stronger -approximation, based on reducing the problem to a matroid secretary
problem on a partition matroid
Improved algorithms and analysis for the laminar matroid secretary problem
In a matroid secretary problem, one is presented with a sequence of objects
of various weights in a random order, and must choose irrevocably to accept or
reject each item. There is a further constraint that the set of items selected
must form an independent set of an associated matroid. Constant-competitive
algorithms (algorithms whose expected solution weight is within a constant
factor of the optimal) are known for many types of matroid secretary problems.
We examine the laminar matroid and show an algorithm achieving provably 0.053
competitive ratio
The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems
We study the matroid secretary problems with submodular valuation functions.
In these problems, the elements arrive in random order. When one element
arrives, we have to make an immediate and irrevocable decision on whether to
accept it or not. The set of accepted elements must form an {\em independent
set} in a predefined matroid. Our objective is to maximize the value of the
accepted elements. In this paper, we focus on the case that the valuation
function is a non-negative and monotonically non-decreasing submodular
function.
We introduce a general algorithm for such {\em submodular matroid secretary
problems}. In particular, we obtain constant competitive algorithms for the
cases of laminar matroids and transversal matroids. Our algorithms can be
further applied to any independent set system defined by the intersection of a
{\em constant} number of laminar matroids, while still achieving constant
competitive ratios. Notice that laminar matroids generalize uniform matroids
and partition matroids.
On the other hand, when the underlying valuation function is linear, our
algorithm achieves a competitive ratio of 9.6 for laminar matroids, which
significantly improves the previous result.Comment: preliminary version appeared in STACS 201
Combinatorial Secretary Problems with Ordinal Information
The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to design online algorithms with small competitive ratios.
For a variety of combinatorial problems, such as bipartite matching, general packing LPs, and independent set with bounded local independence number, we design new algorithms that obtain constant competitive ratios.
For the matroid secretary problem, we observe that many existing algorithms for special matroid structures maintain their competitive ratios even in the ordinal model. In these cases, the restriction to ordinal information does not represent any additional obstacle. Moreover, we show that ordinal variants of the submodular matroid secretary problems can be solved using algorithms for the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we provide a lower bound of Omega(sqrt(n)/log(n)) for algorithms that are oblivious to the matroid structure, where n is the total number of elements. This contrasts an upper bound of O(log n) in the cardinal model, and it shows that the technique of thresholding is not sufficient for good algorithms in the ordinal model
Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information
Random order online contention resolution schemes (ROCRS) are structured
online rounding algorithms with numerous applications and links to other
well-known online selection problems, like the matroid secretary conjecture. We
are interested in ROCRS subject to a matroid constraint, which is among the
most studied constraint families. Previous ROCRS required to know upfront the
full fractional point to be rounded as well as the matroid. It is unclear to
what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang
(SOSA 2022) shed some light on this question by proving that no strong
(constant-selectable) online or even offline contention resolution scheme
exists if the fractional point is unknown, not even for graphic matroids.
In contrast, we show, in a setting with slightly more knowledge and where the
fractional point reveals one by one, that there is hope to obtain strong ROCRS
by providing a simple constant-selectable ROCRS for graphic matroids that only
requires to know the size of the ground set in advance. Moreover, our procedure
holds in the more general adversarial order with a sample setting, where, after
sampling a random constant fraction of the elements, all remaining
(non-sampled) elements may come in adversarial order.Comment: To be published in SOSA2
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class of matroids
representable over a prime field, there exists a constant-competitive matroid
secretary algorithm for the matroids in . This result relies on the
extremely powerful matroid minor structure theory being developed by Geelen,
Gerards and Whittle.
We also note that for asymptotically almost all matroids, the matroid
secretary algorithm that selects a random basis, ignoring weights, is
-competitive. In fact, assuming the conjecture that almost all
matroids are paving, there is a -competitive algorithm for almost all
matroids.Comment: 15 pages, 0 figure
Marches Agricoles: 8/1964 = Agricultural Markets: 8/1964.
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives