Besides the obvious problem of finding a constant-competitive algorithm for general matroids in the adversarial-assignment random-order model there are several other problems to work on. Here is a list of some of them.
Consider the following setting. A weighted matroid with arbitrary hidden weights is presented to an algorithm. At every step, the algorithm can pick any element, ask for its weight, and depending on the answer add it to the solution set or reject it, provided that the solution set is always an independent set. Does this ability of choosing the order help to attain a constant-competitive algorithm for adversarial- value models of the matroid secretary problem?
Another interesting question is to determine if there is an approximate notion of principal partition for domains extending matroids, in particular for matroid intersec- tion domains. An affirmative answer to this question could allow us to use the divide and conquer approach presented for matroids to give low competitive algorithms for the random-assignment random-order model of the corresponding generalized secre- tary problem.
Babaioff et al. [8] (see Lemma 1.4.1) have shown that for general domains, it is impossible to achieve a competitive ratio of o(log n/ log log n). The proof of this lemma shows that this is true even for cases where (i) each element receives their weights from a known distribution, (ii) the algorithm is value-based and (iii) the algorithm can choose the order in which the elements are revealed. Currently, there are no known algorithms achieving an o(n)-competitive ratio for general domains even if the three conditions above hold. It would be interesting to close the gap between the upper and lower bounds on the competitive ratio for this problem.
Finally, it is open to determine if there are cases of the adversarial-assignment model of the generalized secretary problem in which value-based algorithms out- perform comparison-based ones. This seems to be the case since comparison-based algorithms are not allowed to compare the weight of sets of elements, but currently no examples are available.
Part II
Jump Number of Two Directional
Orthogonal Ray Graphs and
Chapter 3
Posets and Perfect Graphs
In this chapter, we define partially ordered sets (posets), comparability graphs and related notions; and state some properties. We then give a survey on different classes of comparability graphs, focusing on subclasses of two directional orthogonal ray graphs and discuss some geometrical characterizations. Finally, we discuss perfect graphs and recall some of their properties.
3.1
Basic Notions of Posets
We mostly follow the notation of Trotter [164].
A partial order relation over a set V is a reflexive, antisymmetric and transitive binary relation. A pair P = (V, ≤P) where ≤P is a partial order over V is called a
partially ordered set or poset.
Two elements u and v in V are comparable if u ≤P v or v ≤P u. Otherwise they
are incomparable. For u and v satisfying u ≤P v, the interval [u, v]P is the set of
elements x such that u ≤P x ≤P v. A linear order, also known as a total order is
a poset in which every pair of elements is comparable. The poset ([n], ≤) where ≤ is the natural order relation is an example of total order.
An element v ∈ V covers u ∈ V if u <P v and there is no w ∈ V such that
u <P v <P w. Two important undirected graphs associated to the poset P are
defined as follows.
The graph GP having V as vertex set such that uv is an edge if u and v are
comparable and u 6= v is the comparability graph of the poset P . A graph G is a comparability graph if there is a poset P such that G is isomorphic to GP.
The graph CP having V as vertex set such that uv is an edge if u covers v or if
v covers u is the covering graph of the poset P . A graph G is a covering graph if
there is a poset P such that G is isomorphic to CP.
Different posets can share the same comparability graph or the same covering graph. Covering graphs are usually represented in the plane with a structured drawing denoted as a Hasse diagram in such a way that if v covers u, then v is above u in the plane. Since the problems considered in this part of the thesis deal with bipartite graphs, the following remark is relevant.
Remark 3.1.1. Every bipartite graph is both a comparability and a covering graph.
Given two posets P = (V, ≤P) and Q = (V, ≤Q) on the same set V , the inter-
section P ∩ Q is defined as the poset (V, ) where u v if and only if u ≤P v and
u ≤Q v. The intersection of a family of posets is defined analogously.
Given two posets P = (U, ≤P) and Q = (V, ≤Q) on possibly different sets, the
product P × Q is defined as the poset (U × V, ), where (u, v) (u0, v0) if u ≤
P u0
and v ≤Q v0. The product of a sequence of posets is defined analogously. For d ≥ 1,
we use Pd to denote the iterative product of d copies of P .
A poset P = (V, ≤P) can be embedded in a poset Q = (W, ≤P) if there is a map
ϕ : V → W for which u ≤P v if and only if ϕ(u) ≤Q ϕ(v). Note that if P can be
embedded in Q and Q can be embedded in R, then the first poset can be embedded in the third one by just composing the corresponding maps.
Unless specifically stated, we assume the set V to be finite and use n to denote its cardinality. The only infinite posets we consider are the usual orders of the integers (Z, ≤), the reals (R, ≤), and their finite powers (Zd, ≤Zd) and (Rd, ≤Rd). We reserve
the symbol ≤, without subindex, to denote the total order between numbers.