The ACGO seminars are held every week as part of an effort of a group of researchers around the topics of Algorithms, Combinatorics, Game Theory and Optimization. Our team consists of researchers in different disciplines and from several chilean universities. If you are interested in giving a talk or receiving the announcements of the seminars, please send us an email to jverschae followed by uc.cl.

**2017-21-06:**

**Author: ** Fabio Botler, DII, U Chile.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, June 21, 14:30.

**Title:** Two conjectures on decompositions of graphs into paths and cycles.

**Abstract:**

A path (resp. cycle) decomposition of a graph G is a set of edge-disjoint paths (resp. cycles) of G that covers the edge-set of G. Gallai (1966) conjectured that every graph on n vertices admits a path decomposition of size at most (n + 1)/2, and Hajós (1968) conjectured that every Eulerian graph on n vertices admits a cycle decomposition of size at most (n − 1)/2. Although these conjectures seem similar, they were verified for distinct classes of graphs. For example, Gallai’s Conjecture was verified for graphs with no cycle composed only by even degree vertices, while Hajós’ Conjecture was verified for graphs with maximum degree at most 4, and for planar graphs. In this talk, I will present a technique that allowed us to verify both conjectures for some (new and old) classes of graphs. We verify both Gallai’s and Hajós’ Conjectures for series–parallel graphs; for graphs with maximum degree 4; and for graphs with treewidth at most 3. Moreover, we show that the only graphs in these classes that do not admit a path decomposition of size at most n/2 are isomorphic to K_3 , K_5 or K_5 – e.

**2017-14-06:**

**Author: ** Sergio Rica, UAI Physics Center, UAI.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, June 14, 14:30.

**Title:** Thermodynamics of small systems through a reversible and conservative discrete automaton.

**Abstract:**

The focus of this talk will be the Q2R model which is a reversible and conservative cellular automaton. The Q2R model possesses quite a rich and complex dynamics. Indeed, the configuration space is composed of a huge number of cycles with exponentially long periods, that we attempt to characterize. Furthermore, a coarse-graining approach is applied to the time series of the total magnetization, leading to a master equation that governs the macroscopic irreversible dynamics of the Q2R automata. The methodology is replicated for various system sizes. In the case of small systems, we show that the master equation leads to a tractable probability transfer matrix of moderate size, which provides a consistent and tractable thermodynamic description. This work is in collaboration with Felipe Urbina and Marco Montalva.

**2017-07-06:**

**Author: ** Pedro Montealegre, UAI.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, June 7, 14:30.

**Title:** Graph Reconstruction in the Congested Clique

**Abstract:**

The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of $n$ nodes. In this talk we consider the situation where the joint input to the nodes is an $n$-node labeled graph $G$, i.e., the local input received by each node is a line of the adjacency matrix of $G$. Nodes execute an algorithm, communicating with each other in synchronous rounds and their goal is to compute some function that depends on $G$.

The most difficult problem we could attempt to solve is the reconstruction problem, where nodes are asked to recover all the edges of the input graph $G$. Formally, given a class of graphs $\mathcal{G}$, the problem is defined as follows: if $G$ is not in $\mathcal{G}$, then every node must reject; on the other hand, if $G$ belongs to $\mathcal{G}$, then every node must end up knowing all the edges of $G$. It is not difficult to see that the total number of bits received by a node through one link is at least $\Omega(\log|\mathcal{G}_n|/n)$, where $\mathcal{G}_n$ is the subclass of all $n$-node labeled graphs in $\mathcal G$.

In this talk we prove that previous bound is tight and that it is possible to achieve it with only two rounds. Moreover, it this bound can be archeived for hereditary classes of graphs in one round. This result recover all known algorithms concerning the reconstruction of graph classes in one round and bandwidth $\cO(\log n)$: forests, planar graphs, cographs, etc. But we also get new one-round algorithms for other hereditary graph classes such as unit disc graphs, interval graphs, etc.

**2017-31-05:**

**Author: ** Diego Arroyuelo, Informatics Department, UTFSM.

**Where:** Republica 779B, Sala P3, 3rd floor..

**When:** Wednesday, May 31, 14:30.

**Title:** Data-aware measures for the dictionary problem.

**Abstract:** Given a sorted set S of size n from a universe U = {1, 2, …, u}, the dictionary problem consists of constructing a data structure for S such that the following queries are supported efficiently: rank (which given a universe element x, counts the number of elements in S that are smaller or equal than x), select (which given a value j, obtains the j-th element in the sorted set S), and member (which test whether a given element belongs to S or not).

Dictionaries are a fundamental building block for many applications in computer science (compressed data structures, information retrieval, text compression), biology (compression of biological sequences), and physical sciences (data collection from particle colliders), among others. Many of these applications are data-intensive, hence the compressed representation of dictionaries becomes crucial.

There exist several compression models for dictionaries, each model defining a compression measure. Such a measure is simply a formula that indicates the minimum amount of bits one could achieve for the representation of set S using that model. In particular, data-aware measures are those that take advantage of the distribution of the n elements in the set to improve compression.

This talk will survey the best known data-aware measures for dictionary compression, as well as the corresponding data structures to support operations rank, select, and member. This will be followed by a discussion about the need of new data-aware measures that achieve better compression in certain circumstances that are of practical relevance. The talk will end proposing a preliminary definition of such a measure, and then a discussion on the consequences of this definition.

**2017-24-05:**

**Author: ** Retsef Levi, MIT Sloan School of Management.

**Where:** Beauchef Poniente (new building), Torre Oriente, 4th Floor, Sala Seminario DII.

**When:** Wednesday, May 24, 14:30.

**Title:** Exploration vs. Exploitation: Reducing Uncertainty in Operational Problems

**Abstract:** Motivated by several core operational applications, we introduce a new class of multistage stochastic optimization models that capture a fundamental tradeoff between performing work and making decisions under uncertainty (exploitation) and investing capacity (and time) to reduce the uncertainty in the decision making (exploration). Unlike existing models, in which the exploration-exploitation tradeoffs typically relate to learning the underlying distributions, the models we introduce assume a known probabilistic characterization of the uncertainty, and focus on the tradeoff of learning (or partially learning) the exact realizations.

For several interesting scheduling models we derive insightful structural results on the optimal policies that lead not only to quantification of the value of learning, but also obtain surprising optimal local decision rules for when it is optimal to explore (learn).

The talk is based on several papers that are joint work with Chen Attias, Tom Magnanti, Robi Krauthgamer and Yaron Shaposhnik.

**2017-17-05:**

**Author: ** Carlos Ochoa, DCC, U Chile.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, May 17, 14:30.

**Title:** Synergistic Solutions on MultiSets

**Abstract:** Karp et al. (1988) described Deferred Data Structures for Multisets as “lazy” data structures which partially sort data to support online rank and select queries, with the minimum amount of work in the worst case over instances of size $n$ and number of queries $q$ fixed. Barbay et al. (2016) refined this approach to take advantage of the gaps between the positions hit by the queries (i.e., the structure in the queries). We develop new techniques in order to further refine this approach and take advantage all at once of the structure (i.e., the multiplicities of the elements), some notions of local order (i.e., the number and sizes of runs) and global order (i.e., the number and positions of existing pivots) in the input; and of the structure and order in the sequence of queries. Our main result is a synergistic deferred data structure which outperforms all solutions in the comparison model that take advantage of only a subset of these features. As intermediate results, we describe two new synergistic sorting algorithms, which take advantage of some notions of structure and order (local and global) in the input, improving upon previous results which take advantage only of the structure (Munro and Spira 1979) or of the local order (Takaoka 1997) in the input; and one new multiselection algorithm which takes advantage of not only the order and structure in the input, but also of the structure in the queries.

**2017-03-05:**

**Author: ** Patricio Poblete, DCC, U Chile.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, May 03, 14:30.

**Title:** Robin Hood Hashing really has constant average search cost and variance in full tables

**Abstract:** Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of $1.883$ when the table was full. Furthermore, by using a non-standard mean-centered search algorithm, this would imply that searches could be performed in expected constant time even in a full table.

In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centered search algorithm runs in expected constant time.

We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by 1/(1-alpha)+O(1), where alpha is the load factor.

To model the behavior of these hash tables, we use a unified approach that we apply also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions.

**2017-26-04:**

**Author: ** Pablo Marquet, PUC.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, April 24, 14:30.

**Title:** What will be the impact of climate change on biodiversity (us included): An open and hot problem.

**2017-12-04:**

**Author: ** Marc Schroder, U Chile.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, April 12, 14:30.

**Title:** Claim games for estate division problems

**Abstract:** The estate division problem, also known as bankruptcy problem, concerns the issue of dividing an estate among a group of claimants when the sum of entitlements exceeds the size of the estate. This problem was formally introduced by O’Neill (1982), after which most of the literature focused on comparing different solution rules by means of their properties. We approach the problem strategically and analyse the claim game that is initiated by O’Neill (1982).

**2017-04-04:**

**Author: ** Jie Han, Universidade de Sao Paulo.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, April 05, 14:30.

**Title:** Perfect Matchings in Hypergraphs

**Abstract:** The problem of determining the minimum d-degree threshold for finding a perfect matching in k-uniform hypergraphs has attracted much attention in the last decade. It is also closely related to the Erdös Matching Conjecture back to 1960s. We will introduce the problem, survey the existing results, and mention some recent progress.

**2017-29-03:**

**Author: ** Kevin Schewior, U Chile.

**Where:** Republica 779B, Sala P3, 3rd floor.

**When:** Wednesday, March 29, 14:30.

**Title:** Chasing Convex Bodies

**Abstract:** We consider the following online problem in d-dimensional Euclidean space. The server initially located in the origin receives an online sequence of convex bodies. In response to each body, the task is to move to a point within that body so as to minimize the total moved distance. We evaluate the performance of online algorithms by competitive analysis. This problem was first considered by Friedman and Linial in 1993 and is an interesting special case of the very general online problem of metrical task systems. Friedman and Linial gave a rather involved constant-competitive algorithm for d=2.

We first look at different greedy policies and notice that they are not constant-competitive even when d=2 and all convex bodies are lines. We then develop a nevertheless simple new algorithm for this special case. Applying a sequence of known and new reductions, we are able to extend this result to a 2^O(d)-competitive algorithm when d is arbitrary and all convex bodies are affine subspaces of the full space. This is the first constant-competitive algorithm in this setting with fixed d>2.

Finally, we discuss directions for future research: Closing the gap between 2^O(d) and the simple lower bound sqrt(d), considering general convex-body chasing, and considering convex-function chasing which is an intermediate problem between convex-body chasing and metrical task systems.

**2017-22-03:**

**Author: ** Krzysztof Fleszar, U Chile.

**Where:** Republica 779B, Sala P3, tercer piso.

**When:** Wednesday, March 22, 14:30.

**Title:** Maximum Disjoint Paths: New Algorithms based on Tree-Likeness

**Abstract:** Maximum Edge Disjoint Paths is a classical NP-hard problem of finding a

maximum-size subset from a given set of k terminal pairs that can be

routed via edge-disjoint paths.

One of the big open problems in approximation algorithms is to close the

gap between the best known approximation upper bound of $\sqrt{n}$

(Chekuri et al. (2006)) and the best known lower bound of $2^{\sqrt{\log

n}}$ (Chuzhoy et al. (2016)). In their seminal paper, Raghavan and

Thompson (Combinatorica, 1987) introduce the technique of randomized

rounding for LPs; their technique gives an O(1)-approximation when edges

may be used by $O(\log n / \log\log n)$ paths.

In this talk, I introduce the problem and present two of our algorithms

(ESA 2016) that strengthen the fundamental results above. They provide

new bounds formulated in terms of the feedback vertex set number r of a

graph, which measures its vertex deletion distance to a forest.

- An $O(\sqrt{r} \log{kr})}$-approximation algorithm. Up to a

logarithmic factor, it strengthens the best known ratio $\sqrt{n}$ due

to Chekuri et al., as $r \le n$.

- An $O(1)$-approximation algorithm with congestion bounded by

$O(\log{kr} / \log\log{kr})$, strengthening the bound obtained by the

classic approach of Raghavan and Thompson.

At the end, an open problem will be stated.

**2017-15-03:**

**Author: ** Saeed Hadikanloo, Univ. de Paris 9.

**Where:** Republica 779B, Sala P3, tercer piso.

**When:** Wednesday, March 15, 14:30.

**Title:** Learning in NonAtomic Anonymous Games: Application to First Order Mean Field Games

**Abstract:**

We introduce a model of anonymous games where the actions are chosen from possibly player dependent sets. We propose several learning procedures similar to the well-known Fictitious Play and Online Mirror Descent and prove their convergence to equilibrium under the classical monotonicity condition. Typical examples are First Order Mean Field Games.

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