2021

Friedemann, Wilhelm; Friedrich, Tobias; Gawendowicz, Hans; Lenzner, Pascal; Melnichenko, Anna; Peters, Jannik; Stephan, Daniel; Vaichenker, Michael Efficiency and Stability in Euclidean Network DesignSymposium on Parallelism in Algorithms and Architectures (SPAA) 2021: 232–242
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Network Design problems typically ask for a minimum cost sub-network from a given host network. This classical point-of-view assumes a central authority enforcing the optimum solution. But how should networks be designed to cope with selfish agents that own parts of the network? In this setting, minimum cost networks may be very unstable in that agents will deviate from a proposed solution if this decreases their individual cost. Hence, designed networks should be both efficient in terms of total cost and stable in terms of the agents' willingness to accept the network. We study this novel type of Network Design problem by investigating the creation of \($(\beta,\gamma)$\)-networks, that are in \($\beta$\)-approximate Nash equilibrium and have a total cost of at most \($\gamma$\) times the optimal cost, for the recently proposed Euclidean Generalized Network Creation Game by Bilò et al.SPAA2019. There, \($n$\) agents corresponding to points in Euclidean space create costly edges among themselves to optimize their centrality in the created network. Our main result is a simple \($\mathcal{O(n^2)$\)-time algorithm that computes a \($(\beta,\beta)$\)-network with low \($\beta$\) for any given set of points. Moreover, on integer grid point sets or random point sets our algorithm achieves a low constant \($\beta$\). Besides these results for the Euclidean model, we discuss a generalization of our algorithm to instances with arbitrary, even non-metric, edge lengths. Moreover, in contrast to these algorithmic results, we show that no such positive results are possible when focusing on either optimal networks, i.e., \($(\beta,1)$\)-networks, or perfectly stable networks, i.e., \($(1,\gamma)$\)-networks, as in both cases NP-hard problems arise, there exist instances with very unstable optimal networks, and there are instances for perfectly stable networks with high total cost. Along the way, we significantly improve several results from Bilò et al. and we asymptotically resolve their conjecture about the Price of Anarchy by providing a tight bound.
@inproceedings{friedemann2021efficiency, abstract = {Network Design problems typically ask for a minimum cost sub-network from a given host network. This classical point-of-view assumes a central authority enforcing the optimum solution. But how should networks be designed to cope with selfish agents that own parts of the network? In this setting, minimum cost networks may be very unstable in that agents will deviate from a proposed solution if this decreases their individual cost. Hence, designed networks should be both efficient in terms of total cost and stable in terms of the agents' willingness to accept the network. We study this novel type of Network Design problem by investigating the creation of $(\beta,\gamma)$-networks, that are in $\beta$-approximate Nash equilibrium and have a total cost of at most $\gamma$ times the optimal cost, for the recently proposed Euclidean Generalized Network Creation Game by Bilò et al.SPAA2019. There, $n$ agents corresponding to points in Euclidean space create costly edges among themselves to optimize their centrality in the created network. Our main result is a simple $\mathcal{O}(n^2)$-time algorithm that computes a $(\beta,\beta)$-network with low $\beta$ for any given set of points. Moreover, on integer grid point sets or random point sets our algorithm achieves a low constant $\beta$. Besides these results for the Euclidean model, we discuss a generalization of our algorithm to instances with arbitrary, even non-metric, edge lengths. Moreover, in contrast to these algorithmic results, we show that no such positive results are possible when focusing on either optimal networks, i.e., $(\beta,1)$-networks, or perfectly stable networks, i.e., $(1,\gamma)$-networks, as in both cases NP-hard problems arise, there exist instances with very unstable optimal networks, and there are instances for perfectly stable networks with high total cost. Along the way, we significantly improve several results from Bilò et al. and we asymptotically resolve their conjecture about the Price of Anarchy by providing a tight bound.}, author = {Friedemann, Wilhelm and Friedrich, Tobias and Gawendowicz, Hans and Lenzner, Pascal and Melnichenko, Anna and Peters, Jannik and Stephan, Daniel and Vaichenker, Michael}, booktitle = {Symposium on Parallelism in Algorithms and Architectures (SPAA)}, keywords = {sys:relevantfor:ae annamelnichenko hansgawendowicz pascallenzner spaa tobiasfriedrich year2021}, pages = {232–242}, publisher = {ASM}, title = {Efficiency and Stability in Euclidean Network Design}, year = 2021 }

2020

Feldmann, Michael; Khazraei, Ardalan; Scheideler, Christian Time- and Space-Optimal Clock Synchronization in the Beeping ModelSymposium Parallelism in Algorithms and Architectures (SPAA) 2020: 223–233
[ Abstract ] [ BibTeX ] [ DOI ]
 
We consider the clock synchronization problem in the (discrete) beeping model: Given a network of \(n\) nodes with each node having a clock value \(\delta(v) \in\) \(\{ 0,\ldots , T-1\}\), the goal is to synchronize the clock values of all nodes such that they have the same value in any round. As is standard in clock synchronization, we assume arbitrary activations for all nodes, i.e., the nodes start their protocol at an arbitrary round (not limited to \(\{0,\ldots,T-1\}\)). We give an asymptotically optimal algorithm that runs in \( 4D + \lfloor D/\lfloor T/4 \rfloor \rfloor \cdot\ \)\((T \mod 4) = O(D)\) rounds, where \(D\) is the diameter of the network. Once all nodes are in sync, they beep at the same round every \(T\) rounds. The algorithm drastically improves on the \(O(T D)\)-bound of [Alistarh et al. 2013] (where \(T\) is required to be at least \(4n\), so the bound is no better than \(O(nD)\)). Our algorithm is very simple as nodes only have to maintain \(3\) bits in addition to the \(\lceil \log T \rceil\) bits needed to maintain the clock. Furthermore we investigate the complexity of self-stabilizing solutions for the clock synchronization problem: We first show lower bounds of \(\Omega(\max\{T,n\})\) rounds on the runtime and \(\Omega(\log(\max\{T,n\}))\) bits of memory required for any such protocol. Afterwards we present a protocol that runs in \(O(\max\{T,n\})\) rounds using at most \(O(\log(\max\{T,n\}))\) bits at each node, which is asymptotically optimal with regards to both, runtime and memory requirements.
@inproceedings{feldmann2020spaceoptimal, abstract = {We consider the clock synchronization problem in the (discrete) beeping model: Given a network of \(n\) nodes with each node having a clock value \(\delta(v) \in\) \(\{ 0,\ldots , T-1\}\), the goal is to synchronize the clock values of all nodes such that they have the same value in any round. As is standard in clock synchronization, we assume arbitrary activations for all nodes, i.e., the nodes start their protocol at an arbitrary round (not limited to \(\{0,\ldots,T-1\}\)). We give an asymptotically optimal algorithm that runs in \( 4D + \lfloor D/\lfloor T/4 \rfloor \rfloor \cdot\ \)\((T \mod 4) = O(D)\) rounds, where \(D\) is the diameter of the network. Once all nodes are in sync, they beep at the same round every \(T\) rounds. The algorithm drastically improves on the \(O(T D)\)-bound of [Alistarh et al. 2013] (where \(T\) is required to be at least \(4n\), so the bound is no better than \(O(nD)\)). Our algorithm is very simple as nodes only have to maintain \(3\) bits in addition to the \(\lceil \log T \rceil\) bits needed to maintain the clock. Furthermore we investigate the complexity of self-stabilizing solutions for the clock synchronization problem: We first show lower bounds of \(\Omega(\max\{T,n\})\) rounds on the runtime and \(\Omega(\log(\max\{T,n\}))\) bits of memory required for any such protocol. Afterwards we present a protocol that runs in \(O(\max\{T,n\})\) rounds using at most \(O(\log(\max\{T,n\}))\) bits at each node, which is asymptotically optimal with regards to both, runtime and memory requirements.}, author = {Feldmann, Michael and Khazraei, Ardalan and Scheideler, Christian}, booktitle = {Symposium Parallelism in Algorithms and Architectures (SPAA)}, keywords = {sys:relevantfor:ae ardalankhazraei christianscheideler michaelfeldmann spaa year2020}, pages = {223-233}, title = {Time- and Space-Optimal Clock Synchronization in the Beeping Model}, year = 2020 }

2019

Bilò, Davide; Friedrich, Tobias; Lenzner, Pascal; Melnichenko, Anna Geometric Network Creation GamesSymposium on Parallelism in Algorithms and Architectures (SPAA) 2019: 323–332
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
Network Creation Games are a well-known approach for explaining and analyzing the structure, quality and dynamics of real-world networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unit-weight edges. In practice, e.g. when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al.~[PODC'03] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state-of-the-art for the unit-weight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight non-constant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the non-metric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of a variant of the classical Network Design Problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.
@inproceedings{bil2019geometric, abstract = {Network Creation Games are a well-known approach for explaining and analyzing the structure, quality and dynamics of real-world networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unit-weight edges. In practice, e.g. when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al. [PODC'03] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state-of-the-art for the unit-weight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight non-constant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the non-metric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of a variant of the classical Network Design Problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.}, author = {Bilò, Davide and Friedrich, Tobias and Lenzner, Pascal and Melnichenko, Anna}, booktitle = {Symposium on Parallelism in Algorithms and Architectures (SPAA)}, keywords = {annamelnichenko davidebilo pascallenzner spaa tobiasfriedrich year2019}, pages = {323-332}, title = {Geometric Network Creation Games}, year = 2019 }

2017

Friedrich, Tobias; Ihde, Sven; Keßler, Christoph; Lenzner, Pascal; Neubert, Stefan; Schumann, David Brief Announcement: Efficient Best Response Computation for Strategic Network Formation under AttackSymposium on Parallelism in Algorithms and Architectures (SPAA) 2017: 321–323
[ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
 
Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NP-hard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by establishing that the recently introduced model by Goyal et al.[WINE'16], which focuses on robust networks in the presence of a strong adversary, is a rare exception which is both realistic and computationally tractable. In particular, we sketch an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure.
@inproceedings{friedrich2017brief, abstract = {Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NP-hard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by establishing that the recently introduced model by Goyal et al.[WINE'16], which focuses on robust networks in the presence of a strong adversary, is a rare exception which is both realistic and computationally tractable. In particular, we sketch an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure.}, author = {Friedrich, Tobias and Ihde, Sven and Keßler, Christoph and Lenzner, Pascal and Neubert, Stefan and Schumann, David}, booktitle = {Symposium on Parallelism in Algorithms and Architectures (SPAA)}, keywords = {christophkessler davidschumann pascallenzner spaa stefanneubert svenihde tobiasfriedrich year2017}, pages = {321-323}, title = {Brief Announcement: Efficient Best Response Computation for Strategic Network Formation under Attack}, year = 2017 }

2013

Kawald, Bernd; Lenzner, Pascal On dynamics in selfish network creationSymposium on Parallelism in Algorithms and Architectures (SPAA) 2013: 83–92
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We consider the dynamic behavior of several variants of the Network Creation Game, introduced by Fabrikant et al. [PODC'03]. Equilibrium networks in these models have desirable properties like low social cost and small diameter, which makes them attractive for the decentralized creation of overlay-networks. Unfortunately, due to the non-constructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequential-move games and analyze if (uncoordinated) selfish play eventually converges to an equilibrium. Thus, we shed light on one of the most natural algorithms for this problem: distributed local search, where in each step some agent performs a myopic selfish improving move. We show that fast convergence is guaranteed for all versions of Swap Games, introduced by Alon et al. [SPAA'10], if the initial network is a tree. Furthermore, we prove that this process can be sped up to an almost optimal number of moves by employing a very natural move policy. Unfortunately, these positive results are no longer true if the initial network has cycles and we show the surprising result that even one non-tree edge suffices to destroy the convergence guarantee. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. Moreover, we show that on non-tree networks no move policy can enforce convergence. We extend our negative results to the well-studied original version, where agents are allowed to buy and delete edges as well. For this model we prove that there is no convergence guarantee - even if all agents play optimally. Even worse, if played on a non-complete host-graph, then there are instances where no sequence of improving moves leads to a stable network. Furthermore, we analyze whether cost-sharing has positive impact on the convergence behavior. For this we consider a version by Corbo and Parkes [PODC'05] where bilateral consent is needed for the creation of an edge and where edge-costs are shared among the involved agents. We show that employing such a cost-sharing rule yields even worse dynamic behavior..
@inproceedings{DBLP:conf/spaa/KawaldL13, abstract = {We consider the dynamic behavior of several variants of the Network Creation Game, introduced by Fabrikant et al. [PODC'03]. Equilibrium networks in these models have desirable properties like low social cost and small diameter, which makes them attractive for the decentralized creation of overlay-networks. Unfortunately, due to the non-constructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequential-move games and analyze if (uncoordinated) selfish play eventually converges to an equilibrium. Thus, we shed light on one of the most natural algorithms for this problem: distributed local search, where in each step some agent performs a myopic selfish improving move. We show that fast convergence is guaranteed for all versions of Swap Games, introduced by Alon et al. [SPAA'10], if the initial network is a tree. Furthermore, we prove that this process can be sped up to an almost optimal number of moves by employing a very natural move policy. Unfortunately, these positive results are no longer true if the initial network has cycles and we show the surprising result that even one non-tree edge suffices to destroy the convergence guarantee. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. Moreover, we show that on non-tree networks no move policy can enforce convergence. We extend our negative results to the well-studied original version, where agents are allowed to buy and delete edges as well. For this model we prove that there is no convergence guarantee - even if all agents play optimally. Even worse, if played on a non-complete host-graph, then there are instances where no sequence of improving moves leads to a stable network. Furthermore, we analyze whether cost-sharing has positive impact on the convergence behavior. For this we consider a version by Corbo and Parkes [PODC'05] where bilateral consent is needed for the creation of an edge and where edge-costs are shared among the involved agents. We show that employing such a cost-sharing rule yields even worse dynamic behavior..}, author = {Kawald, Bernd and Lenzner, Pascal}, booktitle = {Symposium on Parallelism in Algorithms and Architectures (SPAA)}, keywords = {imported pascallenzner spaa year2013}, pages = {83-92}, title = {On dynamics in selfish network creation}, year = 2013 }