The secretary problem or game of Googol are classic models for online selection problems. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given $n$ cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on $n$ consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives: maximizing the probability of selecting the maximum hidden value, and maximizing the expectation of the selected hidden value. For the former objective we obtain a simple $0.45292$-competitive algorithm. For the latter, we obtain a $0.63518$-competitive algorithm. Our main contribution is to set up a model allowing to transform probabilistic optimal stopping problems into purely combinatorial ones. For instance, we can apply our results to obtain lower bounds for the single sample prophet secretary problem.

The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary nonnegative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card. We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value. Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card. We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 – 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples.

Online models that allow recourse can be highly effective in situations where classical online models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple (1.7 + ε)-competitive algorithm using a migration factor of O(1/ε), which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved (4/3+ε)-competitive algorithm using a migration factor of Ō (1/ε 3 ). At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base.

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.

The following zero-sum game is played on a weighted graph G: Alice selects a matching M in G and Bob selects a number k. Then, Alice receives a payoff equal to the ratio of the weight of the top k edges of M to optk, which is the maximum weight of a matching of size at most k in G. If M guarantees a payoff of at least α then it is called α-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a -robust matching, which is best possible for this setting. In this paper, we show that Alice can improve on the guarantee of when allowing her to play a randomized strategy. For this setting, we devise a simple algorithm that returns a 1/ln(4)-robust randomized matching. The algorithm is based on the following non-trivial observation: If all edge weights are integer powers of 2, then any lexicographically optimum matching is 1-robust. We prove this property not only for matchings but for any independence system in which optk is a concave function of k. This class of systems includes matroid intersection, b-matchings, and strong 2-exchange systems. We also show that our robustness results for randomized matchings translate to an asymptotic robustness guarantee for deterministic matchings: When restricting Bob's choice to cardinalities larger than a given constant, then Alice can find a single deterministic matching with approximately the same guaranteed payoff as in the randomized setting. In addition to the above results, we also give a new simple LP-based proof of Hassin and Rubinstein's original result.

After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65--77] proved that any 2-connected graph whose $n$ vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3-1/61236)n, implying that cubic 2-connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller $\epsilon$, we show that the integrality gap of the TSP relaxation is at most $4/3- \epsilon$ even if the graph is not 2-connected (i.e., for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 - 1/18)n.

Trevisan [SIAM J. Comput., 41 (2012), pp. 1769--1786] presented an algorithm for Max-Cut based on spectral partitioning techniques. This is the first algorithm for Max-Cut with an approximation guarantee strictly larger than 1/2 that is not based on semidefinite programming. Trevisan showed that its approximation ratio is of at least 0.531. In this paper we improve this bound up to 0.614247. We also define and extend this result for the more general maximum colored cut problem.

The matroid secretary problem admits several variants according to the order in which the matroid's elements are presented and how the elements are assigned weights. As the main result of this article, we devise the first constant competitive algorithm for the model in which both the order and the weight assignment are selected uniformly at random, achieving a competitive ratio of approximately $5.7187$. This result is based on the nontrivial fact that every matroid can be approximately decomposed into uniformly dense minors. Based on a preliminary version of this work, Oveis Gharan and Vondrák [Proceedings of the 19th Annual European Symposium on Algorithms, ESA, 2011, pp. 335--346] devised a $40e/(e-1)$-competitive algorithm for the stronger random-assignment adversarial-order model. In this article we present an alternative algorithm achieving a competitive ratio of $16e/(e-1)$. As additional results, we obtain new algorithms for the standard model of the matroid secretary problem: the adversarial-assignment random-order model. We present an $O(\log r)$-competitive algorithm for general matroids which, unlike previous ones, uses only comparisons among seen elements. We also present constant competitive algorithms for various matroid classes, such as column-sparse representable matroids and low-density matroids. The latter class includes, as a special case, the duals of graphic matroids.

We present an efficient algorithm to find nonempty minimizers of a symmetric submodular function $f$ over any family of sets ${\cal I}$ closed under inclusion. Our algorithm makes $O(n^3)$ oracle calls to $f$ and ${\cal I}$, where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ [Z. Svitkina and L. Fleischer, in Proceedings of the $49$th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Washington, DC, 2008, pp. 697--706]. We also present two extensions of the above algorithm. The first extension reports all nontrivial inclusionwise minimal minimizers of $f$ over ${\cal I}$ using $O(n^3)$ oracle calls, and the second reports all extreme subsets of $f$ using $O(n^4)$ oracle calls. Our algorithms are similar to a procedure by Nagamochi and Ibaraki [Inform. Process. Lett., 67 (1998), pp. 239--244] that finds all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of size $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [M. Queyranne, Math. Program., 82 (1998), pp. 3--12] to minimize a symmetric submodular function by finding pendent pairs. Our results extend to any class of functions for which we can find a pendent pair whose head is not a given element.

In the Matroid Secretary Problem, introduced by Babaioff et al. [5], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it. The objective is to return a maximum weight independent set of the matroid. There are different variants for this problem depending on the information known about the weights beforehand. In the random assignment model, a hidden list of weights is randomly assigned to the matroid ground set, independently from the random order they are revealed to the algorithm. Our main result is the first constant competitive algorithm for this version of the problem, solving an open question of Babaioff et al. Our algorithm achieves a competitive ratio of 2e2/(e − 1). It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a 2e-competitive algorithm for uniformly dense matroids we also develop. We also present constant competitive algorithms in the standard model where the weights are assigned adversarially, for various classes of matroids including cographic, low density, k-column sparse linear matroids and the case when every element is in a small cocircuit. In the same model, we give a new O(log r)-competitive algorithm for matroids of rank r which only uses the relative order of the weights seen and not their actual values, as previously needed.