Andrés Cristi

In a Stackelberg network pricing game, a leader sets prices for a given subset of edges so as to maximize profit, after which one or multiple followers choose a shortest path from their source to sink. We study the counter-intuitive phenomenon that the use of negative prices by the leader may in fact increase his profit. In doing so, we answer the following two questions. First, how much more profit can the leader earn by setting negative prices? Second, for which network topologies can the profit be increased by using negative prices? Our main result shows that the profit with negative prices can be a factor Θ( (m·k̅)) larger than the maximum profit with positive prices, where m is the number of priceable edges in the graph and k̅≤ 2^m the number of followers. In particular, this factor cannot be bounded for the single follower case, and can even grow linearly in m if the number of followers is large. Our second result shows that series-parallel graphs with a single follower and St...

Two fundamental models in online decision making are that of competitive analysis and that of optimal stopping. In the former the input is produced by an adversary, while in the latter the algorithm has full distributional knowledge of the input. In recent years, there has been a lot of interest in bridging these two models by considering data-driven or sample-based versions of optimal stopping problems. In this paper, we study such a version of the classic single selection optimal stopping problem, as introduced by Kaplan et al. [2020]. In this problem a collection of arbitrary non-negative numbers is shuffled in uniform random order. A decision maker gets to observe a fraction p∈ [0,1) of the numbers and the remaining are revealed sequentially. Upon seeing a number, she must decide whether to take that number and stop the sequence, or to drop it and continue with the next number. Her goal is to maximize the expected value with which she stops. On one end of the spectrum, when p=0,...

In a Stackelberg network pricing game, a leader sets prices for a given subset of edges so as to maximize profit, after which one or multiple followers choose a shortest path from their source to sink. We study the counter-intuitive phenomenon that the use of negative prices by the leader may in fact increase his profit. In doing so, we answer the following two questions. First, how much more profit can the leader earn by setting negative prices? Second, for which network topologies can the profit be increased by using negative prices? Our main result shows that the profit with negative prices can be a factor $\Theta(\log (m\cdot\bar k))$ larger than the maximum profit with positive prices, where $m$ is the number of priceable edges in the graph and $\bar k \leq 2^m$ the number of followers. In particular, this factor cannot be bounded for the single follower case, and can even grow linearly in $m$ if the number of followers is large. Our second result shows that series-parallel gra...

In the secretary problem we are faced with an online sequence of elements with values. Upon seeing an element we have to make an irrevocable take-it-or-leave-it decision. The goal is to maximize the probability of picking the element of maximum value. The most classic version of the problem is that in which the elements arrive in random order and their values are arbitrary. However, by varying the available information, new interesting problems arise. Also the case in which the arrival order is adversarial instead of random leads to interesting variants that have been considered in the literature. In this paper we study both the random order and adversarial order secretary problems with an additional twist. The values are arbitrary, but before starting the online sequence we independently sample each element with a fixed probability $p$. The sampled elements become our information or history set and the game is played over the remaining elements. We call these problems the random or...

The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem [Bar-Noy et al., STOC 2000] and also a QPTAS [Höhn et al., ICALP 2014]. In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slightly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of f(k) · n (1) that outputs a solution with at most (1 + )k tasks or assert that there is no solution...

We present a general model for the operation of a cloud computing server comprised of one or more speed-scalable processors. Typically, tasks are submitted to such a cloud computing server in an online fashion, and the server operator has to schedule the tasks and decides on payments without knowledge about the tasks arriving in the future. Although very natural, this cloud computing problem on speed-scalable processors has not been studied from a mechanism design perspective in the online setting. We provide a mechanism for this setting, both for a single and multiprocessor environment, that has several desirable properties: (1) the induced game admits a subgame perfect equilibrium in pure strategies and therefore a pure Nash equilibrium, (2) the Price of Anarchy is constant, (3) the mechanism is budget balanced, i.e., the sum of the payments of the agents is equal to the total energy costs, (4) the communication complexity is low, (5) the mechanism is computationally tractable for...

With the increased popularity of cloud computing it is of paramount importance to understand energy-efficiency from a game-theoretic perspective. An important question is how the operator of a server should deal with combining energy-efficiency and the particular interests of the users. Consider a cloud server, where clients/agents can submit jobs for processing. The quality of service that each agent perceives is given by a non-decreasing function of the completion time of her job which is private information. The server has to process the jobs and charge each agent while trying to optimize the social cost, defined as the energy expenditure plus the sum of the values of the cost functions of the agents. The operator would like to design a mechanism in order to optimize this objective, which ideally is computationally tractable, charges the users “fairly” and induces a game with an equilibrium.

In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0,1]^2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p in P. In the most prominent variant - Lower-Left-Anchored Rectangle Packing (LLARP) - rectangles are anchored in their lower-left corner. Freedman [W. T. Tutte (Ed.), 1969] conjectured in 1969 that, if (0,0) in P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [Dumitrescu and Toth, 2015]. Maybe even more surprisingly, it is not known whether LLARP - or any ARP-problem with only one anchor - is NP-hard. In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies i...

In the Unsplittable Flow on a Path Cover (UFP-cover) problem we are given a path with a demand for each edge and a set of tasks where each task is defined by a subpath, a size and a cost. The goal is to select a subset of the tasks of minimum cost that together cover the demand of each edge. This problem models various resource allocation settings and also the general caching problem. The best known polynomial time approximation ratio for it is 4 [Bar-Noy et al., STOC 2000]. In this paper, we study the resource augmentation setting in which we need to cover only a slightly smaller demand on each edge than the compared optimal solution. If the cost of each task equals its size (which represents the natural bit-model in the related general caching problem) we provide a polynomial time algorithm that computes a solution of optimal cost. We extend this result to general caching and to the packing version of Unsplittable Flow on a Path in their respective natural resource augmentation se...

The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem (Bar-Noy et al. STOC 2001) and also a QPTAS (Hohn et al. ICALP 2018). In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slighly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of $f(k)\cdot n^{O_{\epsilon }(1)}$ that outputs a solution with at most (1 + 𝜖)k tasks or asserts that t...

We consider Superset, a lesser-known yet interesting variant of the famous card game Set. Here, players look for Supersets instead of Sets, that is, the symmetric difference of two Sets that intersect in exactly one card. In this paper, we pose questions that have been previously posed for Set and provide answers to them; we also show relations between Set and Superset. For the regular Set deck, which can be identified with F^3_4, we give a proof for the fact that the maximum number of cards that can be on the table without having a Superset is 9. This solves an open question posed by McMahon et al. in 2016. For the deck corresponding to F^3_d, we show that this number is Omega(1.442^d) and O(1.733^d). We also compute probabilities of the presence of a superset in a collection of cards drawn uniformly at random. Finally, we consider the computational complexity of deciding whether a multi-value version of Set or Superset is contained in a given set of cards, and show an FPT-reductio...