Jun Kiniwa

Approximation of self-stabilizing vertex cover less than 2

A vertex cover of a graph is a subset of vertices such that each edge has at least one endpoint in the subset. Determining the minimum vertex cover is a well-known NP-complete problem in a sequential setting. Several techniques, e.g., depth-first search, a local ratio theorem, and semidefinite relaxation, have given good approximation algorithms. However, some of them cannot be applied to a distributed setting, in particular self-stabilizing algorithms. Thus only a 2-approximation solution based on a self-stabilizing maximal matching has been obviously known until now. In this paper we propose a new self-stabilizing vertex cover algorithm that achieves (2–1/Δ)-approximation ratio, where Δ is the maximum degree of a given network. We first introduce a sequential (2–1/Δ)-approximation algorithm that uses a maximal matching with the high-degree-first order of vertices. Then we present a self-stabilizing algorithm based on the same idea, and show that the output of the algorithm is the same as that of the sequential one.

Analysis of an Intentional Fault Which Is Undetectable by Local Checks under an Unfair Scheduler

We consider a malicious unfair adversary which generates an undetectable fault by local checks, called an intentional fault. Though the possibility of such a fault has ever been suggested, details of its influence and handling are unknown. We assume the intentional fault in a self-stabilizing mutual exclusion protocol, a hybrid of previously proposed ones that complement each other. In the hybrid protocol, we can cope with the fault by using optional strategies, whether or not sending a minor token, which plays a role of preventing the contamination from spreading. We construct a payoff matrix between a group of privileged processes and an adversary, and consider a multistage two-person zero sum game. We interpret the game in two ways: whether it continues or replays the game after an ME(mutual exclusion)-violating repair, in which more than one unexpected privileges are given. For each case, we evaluate the ability of malicious unfair adversary by using a mixed strategy. Our idea is also considered as a general framework for strengthening an algorithm against an intentional fault.

Request-based token passing for self-stabilizing mutual exclusion

This paper presents a new method for request-based self-stabilizing token passing. A token is passed via a dynamic BFS (breadth-first search) tree rooted at a requesting process. When one of the tree edges reaches a process that has a token, the process is aware of the occurrence of a request. Then the token is passed towards the root. Even if multiple tokens stay at distinct roots, it can be shown that they will be merged into a single token. Furthermore, each request-rooted tree continues to grow until the request is serviced. As a result, the tree with a request that has long been neglected grows larger and larger, which makes it easier for the requesting process to get a token. Such advantages can be achieved by using bounded memory. We also evaluate the stabilization time and the efficiency of servicing k requests, called k-covering time, of our method.

Price stabilization in networks: what is an appropriate model?

We consider a simple network model for economic agents where each can buy commodities in the neighborhood. Their prices may be initially distinct in any node. However, by assuming some rules on new prices, we show that the distinct prices will converge to unique by iterating buy and sell operations. First, we present a protocol model in which each agent always bids an arbitrary price in the difference between his own price and the lowest price in the neighborhood, called max price difference. Next, we derive the condition that price stabilization occurs in our model. Furthermore, we consider game (auction) theoretic price determination by assuming that each agents value is uniformly distributed over the max price difference. Finally, we perform a simulation experiment. Our model is suitable for investigating the effects of network topologies on price stabilization.

An Inflation / Deflation Model for Price Stabilization in Networks

We consider a simple network model for economic agents where each can buy goods in the neighborhood. Their prices may be initially distinct in any node. However, by assuming some rules on new prices, we show that the distinct prices will reach an equilibrium price by iterating buy and sell operations. First, we present a protocol model in which each agent always bids at some rate in the difference between his own price and the lowest price in the neighborhood. Next, we show that the equilibrium price can be derived from the total funds and the total goods for any network. This confirms that the inflation / deflation occurs due to the increment / decrement of funds as long as the quantity of goods is constant. Finally, we consider how injected funds spread in a path network because sufficient funds of each agent drive him to buy goods. This is a monetary policy for deflation. A set of recurrences lead to the price of goods at each node at any time. Then, we compare two injections with half funds and single injection. It turns out the former is better than the latter from a fund-spreading point of view, and thus it has an application to a monetary policy and a strategic management based on the information of each agent.

Effective Search for a Naval Mine with Application to Distributed Failure Detection

We consider a kind of reconnaissance problem which has an application to distributed failure detection. The problem can be considered as a multistage two-person zero-sum game. The two-person, player A and player B, consists of a transport ship and a terrorist, respectively, where the ship is equipped with an unmanned reconnaissance boat. The ship circulates ports again and again and the terrorist may lay a naval mine on the shipping route. For safety, the ship dispatches the unmanned reconnaissance boat and removes the risk of a mine. However, it is very rare that the terrorist lays the mine, while the circulation of the reconnaissance boat is very costly. So, we introduce a mine-preparing probability, represented by geometric distribution, preceding the terrorist’s strategy. The ship has to determine when it should dispatch the boat so that it can maximize its expected payoff. First, we assume that the mine is laid at each beginning of a stage and investigate two cases, a game continuation case and a game termination case, after the ship has been broken by a mine. Next, we assume that the mine may be laid at any timing of a stage and investigate two methods, dispatching two boats and dispatching one boat, for the game continuation case. Finally, we state that the problem can also be applied to a failure detection problem in a distributed system if we regard the ship as a token and the terrorist as an adversary who causes a failure.

Asset Value Game and Its Extension: Taking Past Actions into Consideration

In 1997, a minority game (MG) was proposed as a non-cooperative iterated game with an odd population of agents who make bids whether to buy or sell. Since then, many variants of the MG have been proposed. However, the common disadvantage in their characteristics is to ignore the past actions beyond a constant memory. So it is difficult to simulate actual payoffs of agents if the past price behavior has a significant influence on the current decision. In this paper we present a new variant of the MG, called an asset value game (AG), and its extension, called an extended asset value game (ExAG). In the AG, since every agent aims to decrease the mean acquisition cost of his asset, he automatically takes the past actions into consideration. The AG, however, is too simple to reproduce the complete market dynamics, that is, there may be some time lag between the price and his action. So we further consider the ExAG by using probabilistic actions, and compare them by simulation.

Brief announcement: an efficient and self-stabilizing link formation algorithm

We propose a self-stabilizing link formation algorithm based on a cooperative network formation game. An underlying network G = (V,E) consists of n processors represented by nodes V = {1,2,...,n}, and communication links represented by edges E = {ij | i,j ∈ V}. An agent network L = (A,E L ), where A = V and E L ⊆E, is defined on G. Let δ i be a benefit that agent i provides others, and c ij a cost of linking i with j that agent i incurs. We assume a state-reading model, a fair distributed daemon, and a token circulation for formation/severance of links.