Ian Post

Liquidity in credit networks: a little trust goes a long way

Credit networks represent a way of modeling trust between entities in a network. Nodes in the network print their own currency and trust each other for a certain amount of each others currency. This allows the network to serve as a decentralized payment infrastructure --- arbitrary payments can be routed through the network by passing IOUs between trusting nodes in their respective currencies --- and obviates the need for a common currency. Credit networks have the property that the loss incurred by nodes in the network due to the presence of malicious nodes is bounded and localized: a node cannot incur a loss greater than the total credit extended by it to other nodes in the network and the only nodes that incur a loss are the ones that extend credit to the malicious nodes. Also, routing payments in credit networks is not significantly less efficient compared to a centralized network since it only requires a max-flow computation. These properties make this model useful not only for monetary transactions, but also in any setting where there is a need to model trust between nodes in a network. It has been shown to be particularly well-suited for transactions in exchange economies such as P2P networks where it can be used to improve inefficiencies resulting from asynchronous demand and bilateral trading. It has been used as a way of imposing group budget constraint on bidders in a multi-unit auction. It can also be used in settings such as packet routing in mobile ad-hoc networks and combating spam in viral marketing over social networks. It has applications in military scenarios where it is important to know who to trust. However, in order for the model to be of practical use it should be able to support repeated transactions between nodes over a long period of time. This motivates the following question which we formulate and study in this paper: if the network is sufficiently well-connected and has sufficient credit to begin with, can we sustain transactions in perpetuity without additional injection of credit? How does liquidity depend upon network topology and transaction rates between nodes, and how does it compare with a centralized currency infrastructure with a common currency? We study these questions under a simple model of repeated transactions: at each time step we pick a pair of nodes (s, t) in the network with probability λst and try to route a unit payment along the shortest feasible path from s to t. If such a path exists, we route the flow and modify edge capacities along the path. The transaction fails if there is no path from s to t. We show that the success probability of transactions is independent of the path used to route flow between nodes. For symmetric transaction rates, we show analytically and via simulations that the success probability for complete graphs, Erdös-Rényi graphs and preferential attachment graphs goes to one as the size, the density or the credit capacity of the network increases. Further, we characterize a centralized currency system as a special type of a star network (one where edges to the root have infinite credit capacity, and transactions occur only between leaf nodes) and compute its steady-state success probability. We show how to construct a centralized system that is equivalent to a given credit network and that the steady-state failure probability in complete graphs and Erdös-Rényi networks is at most a constant-factor worse than equivalent centralized currency systems. So in return for all the benefits resulting from the decentralized nature of the system, we do not give up a lot of liquidity compared to a centralized model with a common currency.

The Simplex Method is Strongly Polynomial for Deterministic Markov Decision Processes

We prove that the simplex method with the highest gain/most-negative-reduced cost pivoting rule converges in strongly polynomial time for deterministic Markov decision processes (MDPs) regardless of the discount factor. For a deterministic MDP with n states and m actions, we prove the simplex method runs in O(n3 m2 log2 n) iterations if the discount factor is uniform and O(n5 m3 log2 n) iterations if each action has a distinct discount factor. Previously the simplex method was known to run in polynomial time only for discounted MDPs where the discount was bounded away from 1.

An Oblivious O(1)-Approximation for Single Source Buy-at-Bulk

We consider the single-source (or single-sink) buy-at-bulk problem with an unknown concave cost function. We want to route a set of demands along a graph to or from a designated root node, and the cost of routing x units of flow along an edge is proportional to some concave, non-decreasing function f such that f(0) = 0. We present a polynomial time algorithm that finds a distribution over trees such that the expected cost of a tree for any f is within an O(1)-factor of the optimum cost for that f. The previous best simultaneous approximation for this problem, even ignoring computation time, was O(log |D|), where D is the multi-set of demand nodes. We design a simple algorithmic framework using the ellipsoid method that finds an O(1)-approximation if one exists, and then construct a separation oracle using a novel adaptation of the Guha, Meyerson, and Munagala[GMM01] algorithm for the single-sink buy-at-bulk problem that proves an O(1) approximation is possible for all f. The number of trees in the support of the distribution constructed by our algorithm is at most 1+log |D|.

One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk

We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)-approximations for all concave functions were previously not known to exist regardless of computation time.

Embedding paths into trees: VM placement to minimize congestion

Modern cloud infrastructure providers allow customers to rent computing capability in the form of a network of virtual machines (VMs) with bandwidth guarantees between pairs of VMs. Typical requests are in the form of a chain of VMs with an uplink bandwidth to the gateway node of the network (rooted path requests), and most data center architectures route network packets along a spanning tree of the physical network. VMs are instantiated inside servers which reside at the leaves of this network, leading to the following optimization problem: given a rooted tree network T and a set of rooted path requests, find an embedding of the requests that minimizes link congestion. Our main result is an algorithm that, given a rooted tree network T with n leaves and set of weighted rooted path requests, embeds a 1−e fraction of the requests with congestion at most poly(logn, logθ,e−1)·OPT (approximation is necessary since the problem is NP-hard). Here OPT is the congestion of the optimal embedding and θ is the ratio of the maximum to minimum weights of the path requests. We also obtain an O(Hlogn/e2) approximation if node capacities can be augmented by a (1+e) factor (here H is the height of the tree). Our algorithm applies a randomized rounding scheme based on Group Steiner Tree rounding to a novel LP relaxation of the set of subtrees of T with a given number of leaves that may be of independent interest.