Aug. 19, 2016, 12:15 p.m.

## [COG seminar]. Marcin Bieńkowski, Adam Kunysz, Krzysztof Sornat: Our ESA'16 & MFCS'16 papers

**Piątek, 12:15-14:00, sala 310.**

**Marcin Bieńkowski: ****Online Algorithms for Multi-Level Aggregation**

**Abstract.** In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests.

Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines.

Joint work with Martin Böhm, Jarosław Byrka, Marek Chrobak, Christoph Dürr, Lukáš Folwarczný, Łukasz Jeż, Jiří Sgall, Nguyen Kim Thang and Pavel Veselý, to appear at ESA 2016.

**Adam Kunysz: The Strongly Stable Roommates Problem**

**Abstract.** An instance of the strongly stable roommates problem with incomplete lists and ties SRTI is an undirected non-bipartite graph G = (V, E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. Ties are disjoint and may contain one vertex. A matching M is a set of vertex-disjoint edges. An edge {x, y} \in E \ M is a *blocking edge* for M if x is either unmatched or strictly prefers y to its current partner in M, and y is either unmatched or strictly prefers x to its current partner in M or is indifferent between them. A matching is *strongly stable* if there is no blocking edge with respect to it. We present an O(nm) time algorithm for computing a strongly stable matching, where we denote n = |V| and m = |E|. The best previously known solution had running time O(m^2). We also give a characterisation of the set of all strongly stable matchings. We show that there exists a partial order with O(m) elements representing the set of all strongly stable matchings, and wegive an O(nm) algorithm for constructing such a representation. Our algorithms are based on a simple reduction to the bipartite version of the problem.

To appear at ESA 2016.

**Krzysztof Sornat: Inequity Aversion Pricing over Social Networks: Approximation Algorithms and Hardness Results**

**Abstract. **We study a revenue maximization problem in the context of social networks. Namely, we consider a model introduced by Alon, Mansour, and Tennenholtz (EC 2013) that captures inequity aversion, i.e., prices offered to neighboring vertices should not be significantly different. We first provide approximation algorithms for a natural class of instances, referred to as the class of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for example, to settings where the seller will only consider a fixed number of discount types or special offers. We then resolve one of the open questions posed in Alon et al., by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be relatively large. Finally, we also provide some extensions of the model of Alon et al., regarding either the allowed set of prices, or the demand type of the clients.

Joint work with G. Amanatidis and E. Markakis, to appear at MFCS 2016.