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We study the problem of routing traffic for independent selfish users in a congested network to minimize the total latency. The inefficiency of selfish routing motivates regulating the flow of the system to lower the total latency of the Nash Equilibrium by economic incentives or penalties. When applying tax to the routes, we follow the definition of [Christodoulou et al, Algorithmica, 2014] to define ePoA as the Nash total cost including tax in the taxed network over the optimal cost in the original network. We propose a simple tax scheme consisting of step functions imposed on the links. The tax scheme can be applied to routing games with parallel links, affine cost functions and single-commodity networks to lower the ePoA to at most 4/3 - epsilon, where epsilon only depends on the discrepancy between the links. We show that there exists a tax scheme in the two link case with an ePoA upperbound less than 1.192 which is almost tight. Moreover, we design another tax scheme that lowers ePoA down to 1.281 for routing games with groups of links such that links in the same group are similar to each other and groups are sufficiently different.

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.

Understanding the algorithmic behaviors that are in principle realizable in a chemical system is necessary for a rigorous understanding of the design principles of biological regulatory networks. Further, advances in synthetic biology herald the time when we'll be able to rationally engineer complex chemical systems, and when idealized formal models will be- come blueprints for engineering. Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here we study the following problem: what functions f : ℝ^k → ℝ can be computed by a chemical reaction network, in which the CRN eventually produces the correct amount of the "output" molecule, no matter the rate at which reactions proceed? This captures a previously unexplored, but very natural class of computations: for example, the reaction X_1 + X_2 → Y can be thought to compute the function y = min(x_1, x_2). Such a CRN is robust in the sense that it is correct whether its evolution is governed by the standard model of mass-action kinetics, alternatives such as Hill-function or Michaelis-Menten kinetics, or other arbitrary models of chemistry that respect the (fundamentally digital) stoichiometric constraints (what are the reactants and products?). We develop a formal definition of such computation using a novel notion of reachability, and prove that a function is computable in this manner if and only if it is continuous piecewise linear.

We study the power of uncontrolled random molecular movement in a model of self-assembly called the nubots model. The nubots model is an asynchronous nondeterministic cellular automaton augmented with rigid-body movement rules (push/pull, deterministically and programmatically applied to specific monomers) and random agitations (nondeterministically applied to every monomer and direction with equal probability all of the time). Previous work on nubots showed how to build simple shapes such as lines and squares quickly—in expected time that is merely logarithmic of their size. These results crucially make use of the programmable rigid-body movement rule: the ability for a single monomer to push or pull large objects quickly, and only at a time and place of the programmers’ choosing. However, in engineered molecular systems, molecular motion is largely uncontrolled and fundamentally random. This raises the question of whether similar results can be achieved in a more restrictive, and perhaps easier to justify, model where uncontrolled random movements, or agitations, are happening throughout the self-assembly process and are the only form of rigid-body movement. We show that this is indeed the case: we give a polylogarithmic expected time construction for squares using agitation, and a sublinear expected time construction to build a line. Such results are impossible in an agitation-free (and movement-free) setting and thus show the benefits of exploiting uncontrolled random movement.

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. Assuming a fixed molecular population size and bimolecular reactions, CRNs are formally equivalent to population protocols, a model of distributed computing introduced by Angluin, Aspnes, Diamadi, Fischer, and Peralta (PODC 2004). The challenge of fast computation by CRNs (or population protocols) is to ensure that there is never a bottleneck "slow" reaction that requires two molecules (agent states) to react (communicate), both of which are present in low (O(1)) counts. It is known that CRNs can be fast in expectation by avoiding slow reactions with high probability. However, states may be reachable (with low probability) from which the correct answer may only be computed by executing a slow reaction. We deem such an event a speed fault. We show that the problems decidable by CRNs guaranteed to avoid speed faults are precisely the detection problems: Boolean combinations of questions of the form "is a certain species present or not?". This implies, for instance, that no speed fault free CRN could decide whether there are at least two molecules of a certain species, although a CRN could decide this in "fast" expected time — i.e. speed fault free CRNs "can't count."

While computer science has given us a framework for determining the complexity and difficulty of solving computational problems, we do not yet have a theoretical framework for knowing what actions, behaviors, and life-like qualities can emerge from a given set of simple modular units. There has been much interest in developing models for programming active self-assembly processes in both the reconfigurable robotics community and the nanotechnology community. With respect to materials science and nanotechnology, the models proposed to date are either not yet implementable with our current understanding of synthetic chemistry or those that are implementable are limited to a set of features that do not capture the power of active components. Prior implementable models of molecular assembly only considered the passive behaviors of attaching and detaching from a complex. Inspired by the algorithmic tile assembly model [Winfree, 1996] and the graph grammar assembly model [Klavins et al., 2004], we describe a formal model for studying the complexity of self-assembled structures with active molecular components. In particular, we add an insertion primitive and we show a direct mapping of our model to a molecular implementation using DNA. We show that the expressive power of this language is stronger than regular languages, but at most as strong as context free grammars. Here, we explore the trade-off between the complexity of the system (in terms of the number of unit types), and the behavior of the system and speed of its assembly. We find that we can grow a line of any given length n in expected time O(log^3 n) using O(log^2 n) monomers. If we grow a line with k insertion rules, either the expected final length is infinite or the expected length at time t is at most (2t+2)^k^2, which is polynomial in t.

Winfree’s abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n ×n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature” (the binding strength threshold required for a tile to attach).

DNA tile self-assembly has emerged as a rich and promising primitive for nano-technology. This paper studies the problems of minimizing assembly time and error rate by changing the tile concentrations because changing the tile concentrations is easy to implement in actual lab experiments. We prove that setting the concentration of tile T_i proportional to the square root of N_i where N_i is the number of times T_i appears outside the seed structure in the final assembled shape minimizes the rate of growth errors for rectilinear tile systems. We also show that t he same concentrations minimize the expected assembly time for a feasible class of tile systems. Moreover, for general tile systems, given tile concentrations, we can approximate the expected assembly time with high accuracy and probability by running only a polynomial number of simulations in the size of the target shape.

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. We address the following synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability synthesis dates back to von Neummann's 1951 work and was followed, among others, by Knuth and Yao in 1976, who demonstrated that arbitrary rational probabilities can be generated with tree networks; where minimizing the expected path length, the expected number of coin tosses in their paradigm, is the key consideration. Motivated by the synthesis of stochastic DNA based molecular systems, we focus on designing optimal-sized stochastic flow networks (the size of a network is the number of splitters). We assume that each splitter has two outgoing edges and is unbiased (probability 1/2 per output edge). We show that an arbitrary rational probability a/b with a ≤ b ≤ 2^n can be realized by a stochastic flow network of size n, we also show that this is optimal. We note that our stochastic flow networks have feedback (cycles in the network), in fact, we demonstrate that feedback improves the expressibility of stochastic flow networks, since without feedback only probabilities of the form ^a/_2^n) (a an integer) can be realized.

Load balancing is a common approach for task assignment in distributed architectures. In this paper, we show that the degree of inefficiency in load balancing designs is highly dependent on the scheduling discipline used at each of the backend servers. Traditionally, the back-end scheduler can be modeled as Processor Sharing (PS), in which case the degree of inefficiency grows linearly with the number of servers. However, if the back-end scheduler is changed to Shortest Remaining Processing Time (SRPT), the degree of inefficiency can be independent of the number of servers, instead depending only on the heterogeneity of the speeds of the servers. Further, switching the back-end scheduler to SRPT can provide significant improvements in the overall mean response time of the system as long as the heterogeneity of the server speeds is small.