Ho-Lin Chen

Network design with weighted players

We consider a model of game-theoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. [2] proved that pure-strategy Nash equilibria always exist and that the price of stability--the ratio in costs of a minimumcost Nash equilibrium and an optimal solution--is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight wi ≥ 1, and its cost share of an edge in its path equals wi times the edge cost, divided by the total weight of the players using the edge.This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria--outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log wmax)-approximate Nash equilibria exist in all weighted Shapley network design games, where wmax is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α = Ω(log wmax), the price of stability with respect to O(α)- approximate Nash equilibria is O((log W)/α), where W is the sum of the players weights. In particular, there is always an O(logW)-approximate Nash equilibrium with cost within a constant factor of optimal.Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log wmax/ log log wmax)-approximate Nash equilibria, and show that for all α = Ω(logwmax/ log log wmax), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.

Designing Network Protocols for Good Equilibria

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs.

Error free self-assembly using error prone tiles

DNA self-assembly is emerging as a key paradigm for nano-technology, nano-computation, and several related disciplines. In nature, DNA self-assembly is often equipped with explicit mechanisms for both error prevention and error correction. For artificial self-assembly, these problems are even more important since we are interested in assembling large systems with great precision. n nWe present an error-correction scheme, called snaked proof-reading, which can correct both growth and nucleation errors in a self-assembling system. This builds upon an earlier construction of Winfree and Bekbolatov [11], which could correct a limited class of growth errors. Like their construction, our system also replaces each tile in the system by a k × k block of tiles, and does not require changing the basic tile assembly model proposed by Rothemund and Winfree [8]. n nWe perform a theoretical analysis of our system under fairly general assumptions: tiles can both attach and fall off depending on the thermodynamic rate parameters which also govern the error rate. We prove that with appropriate values of the block size, a seed row of n tiles can be extended into an n × n square of tiles without errors in expected time

Network Design with Weighted Players

widetilde{O}(n)

Active self-assembly of algorithmic shapes and patterns in polylogarithmic time

, and further, this square remains stable for an expected time of

Designing networks with good equilibria

widetilde{Omega}(n)

Reducing facet nucleation during algorithmic self-assembly

. This is the first error-correction system for DNA self-assembly that has provably good assembly time (close to linear) and provable error-correction. The assembly time is thesame, up to logarithmic factors, as the time for an irreversible, error-free assembly. n nWe also did a preliminary simulation study of our scheme. In simulations, our scheme performs much better (in terms of error-correction) than the earlier scheme of Winfree and Bekbolatov, and also much better than the unaltered tile system. n nOur basic construction (and analysis) applies to all rectilinear tile systems (where growth happens from south to north and west to east). These systems include the Sierpinski tile system, the square-completion tile system, and the block cellular automata for simulating Turing machines. It also applies to counters, a basic primitive in many self-assembly constructions and computations.

Invadable self-assembly: combining robustness with efficiency

We consider a model of game-theoretic network design initially studied by Anshelevich etxa0al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004), where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich etxa0al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004) proved that pure-strategy Nash equilibria always exist and that the price of stability—the ratio between the cost of the best Nash equilibrium and that of an optimal solution—is Θ(logu2009k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight wi≥1, and its cost share of an edge in its path equals wi times the edge cost, divided by the total weight of the players using the edge.This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria—outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(logu2009wmaxu2009)-approximate Nash equilibria exist in all weighted Shapley network design games, where wmaxu2009 is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α=Ω(logu2009wmaxu2009), the price of stability with respect to O(α)-approximate Nash equilibria is O((logu2009W)/α), where W is the sum of the players’ weights. In particular, there is always an O(logu2009W)-approximate Nash equilibrium with cost within a constant factor of optimal.Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(logu2009wmaxu2009/ logu2009logu2009wmaxu2009)-approximate Nash equilibria, and show that for all α=Ω(logu2009wmaxu2009/logu2009logu2009wmaxu2009), achieving a price of stability of O(logu2009W/α) requires relaxing equilibrium constraints by an Ω(α) factor.

Dimension augmentation and combinatorial criteria for efficient error-resistant DNA self-assembly

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biologys fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time polylogarithmic in the size of the shape plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components.