Topics in algorithmic, enumerative and geometric combinatorics

This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S_2xS_{n-2}-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai's 3^d-conjecture for such polytopes. Furthermore, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given. We end by constructing an interesting class of Hansen polytopes having very few faces and yet not being Hanner. The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the packing density for some classes of generalized patterns, including all the three letter patterns. In the fourth paper, we present combinatorial proofs of the enumeration of derangements with descents in prescribed positions. To this end, we consider fixed point lambda-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n. The fifth paper solves a partially ordered generalization of the famous secretary problem. The elements of a finite nonempty partially ordered set are exposed in uniform random order to a selector who, at any given time, can see the relative order of the exposed elements. The selector's task is to choose online a maximal element. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order, thus proving that the linearly ordered set is at least as difficult as any other instance of the problem. To this end, we define a probability measure on the maximal elements of an arbitrary partially ordered set, that may be interesting in its own right

What Do Our Choices Say About Our Preferences?

Taking online decisions is a part of everyday life. Think of buying a house, parking a car or taking part in an auction. We often take those decisions publicly, which may breach our privacy - a party observing our choices may learn a lot about our preferences. In this paper we investigate the online stopping algorithms from the privacy preserving perspective, using a mathematically rigorous differential privacy notion. In differentially private algorithms there is usually an issue of balancing the privacy and utility. In this regime, in most cases, having both optimality and high level of privacy at the same time is impossible. We propose a natural mechanism to achieve a controllable trade-off, quantified by a parameter, between the accuracy of the online algorithm and its privacy. Depending on the parameter, our mechanism can be optimal with weaker differential privacy or suboptimal, yet more privacy-preserving. We conduct a detailed accuracy and privacy analysis of our mechanism applied to the optimal algorithm for the classical secretary problem. Thereby the classical notions from two distinct areas - optimal stopping and differential privacy - meet for the first time.Comment: 22 pages, 6 figure

Counting embeddings of rooted trees into families of rooted trees

The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in which the maximal elements of $S$ and $T$ overlap. We investigate the number of good and all embeddings of a rooted poset $S$ in the family of all binary trees on $n$ elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset $S$ in the family of all planted plane trees of size $n$. We derive the asymptotic behaviour of good and all embeddings in all cases and we prove that the ratio of good embeddings to all is of the order $\Theta(1/\sqrt{n})$ in all cases, where we provide the exact constants. Furthermore, we show that this ratio is non-decreasing with $S$ in the plane binary case and asymptotically non-decreasing with $S$ in the non-plane binary case and in the planted plane case. Finally, we comment on the case when $S$ is disconnected.Comment: 20 pages, 6 figure

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

The Best-or-Worst and the Postdoc problems

We consider two variants of the secretary problem, the\emph{ Best-or-Worst} and the \emph{Postdoc} problems, which are closely related. First, we prove that both variants, in their standard form with binary payoff 1 or 0, share the same optimal stopping rule. We also consider additional cost/perquisites depending on the number of interviewed candidates. In these situations the optimal strategies are very different. Finally, we also focus on the Best-or-Worst variant with different payments depending on whether the selected candidate is the best or the worst

Problems of optimal choice on posets and generalizations of acyclic colourings

NOTE : The mathematical symbols in the abstract do not always display correctly in this text field. Please see the abstract in the thesis for the definitive abstract. ABSTRACT: This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the `secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree. I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove. In Chapter 2, I shall suppose that the candidates come as ‘m’ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m → ∞. I shall also find an optimal strategy and its probability of success for the analagous version with ‘c’-tuplets. I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least ¹/e . For posets with ‘k ≥ 2’ maximal elements, I shall prove that if their width is also ‘k’ then this can be improved to ‘k-1√1/k’ and show that no better bound of this type is possible. In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form. I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least ’l’ must receive at least three colours. In the second, every cycle must receive at least ‘c’ colours, except those of length less than ‘c’, which must be multicoloured. My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ‘r’ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D