Relaxation and Metastability in the RandomWalkSAT search procedure

An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte

Polynomial iterative algorithms for coloring and analyzing random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This lead us to propose a new algorithm able to color in polynomial time random graphs in the hard but colorable region, i.e when $c\in [c_d,c_q]$.Comment: 23 pages, 10 eps figure

Locating information with uncertainty in fully interconnected networks with applications to World Wide Web information retrieval

In this paper we examine the problem of searching for some information item in the nodes of a fully interconnected computer network, where each node contains information relevant to some topic as well as links to other network nodes that also contain information, not necessarily related to locally kept information. These links are used to facilitate the Internet users and mobile software agents that try to locate specific pieces of information. However, the links do not necessarily point to nodes containing information of interest to the user or relevant to the aims of the mobile agent. Thus an element of uncertainty is introduced. For example, when an Internet user or some search agent lands on a particular network node, they see a set of links that point to information that is, supposedly, relevant to the current search. Therefore, we can assume that a link points to relevant information with some unknown probability p that, in general, is related to the number of nodes in the network (intuitively, as the network grows, this probability tends to zero since adding more nodes to the network renders some extant

Stackelberg strategies and collusion in network games with splittable flow

We study the impact of collusion in network games with splittable flow and focus on the well established price of anarchy as a measure of this impact. We first investigate symmetric load balancing games and show that the price of anarchy is bounded from above by m, where m denotes the number of coalitions. For general networks, we present an instance showing that the price of anarchy is unbounded, even in the case of two coalitions. If latencies are restricted to polynomials, we prove upper bounds on the price of anarchy for general networks, which improve upon the current best ones except for affine latencies. In light of the negative results even for two coalitions, we analyze the effectiveness of Stack-elberg strategies as a means to improve the quality of Nash equilibria. In this setting, an α fraction of the entire demand is first routed centrally by a Stackelberg leader according to a pre-defined Stackelberg strategy and the remaining demand is then routed selfishly by the coalitions (followers). For a single coalitional follower and parallel arcs, we develop an efficiently computable Stackelberg strategy that reduces the price of anarchy to one. For general networks and a single coalitional follower, we show that a simple strategy, called SCALE, reduces the price of anarchy to 1+α. Finally, we investigate SCALE for multiple coalitional followers, general networks, and affine latencies. We present the first known upper bound on the price of anarchy in this case. Our bound smoothly varies between 1.5 when α = 0 and full efficiency when α = 1.

Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners

In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the \Theta_2^p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates--although the number of voters may still be polynomial in the number of candidates--a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it ``knows'' that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner