CAPITULO V PLANEAMIENTO INTEGRAL
Jueves 8 de junio de 2006 NORMAS LEGALES 320507 (c) Cuando sean previstas residencias para el personal,
2005; Fotakis, Kontogiannis, Spirakis PAULSPIRAKIS
Computer Engineering and Informatics, Research and Academic Computer Technology Institute, Patras University, Patras, Greece
Keywords and Synonyms Atomic selfish flows
Problem Definition
A setting is assumed in whichnselfish users compete for routing their loads in a network. The network is anst
directed graph with a single source vertexsand a single destination vertext. The users are ordered sequentially. It is assumed that each user plays after the user before her in the ordering, and the desired end result is a Pure Nash Equilibrium (PNEfor short). It is assumed that, when a user plays (i. e. when she selects anst path to route her load), the play is a best response (i. e. minimum de- lay), given the paths and loads of users currently in the net. The problem then is to find the class of directed graphs for which such an ordering exists so that the implied sequence of best responses leads indeed to a Pure Nash Equilibrium.
The Model
Anetwork congestion gameis a tuple ((wi)i2N;G;(de)e2E)
whereN=f1; : : : ;ngis the set of users where usericon- trols wi units of traffic demand. In unweighted conges-
tion games wi= 1 for i= 1; : : : ;n. G(V,E) is a directed
graph representing the communications network andde
is the latency function associated with edgee2E. It is as- sumed that thede’s are non-negative and non-decreasing
functions of the edge loads. The edges are calledidenti- califde(x) =x; 8e2E. The model is further restricted
to single-commodity network congestion games, whereG
has a single sourcesand destinationtand the set of users’ strategies is the set ofstpaths, denotedP. Without loss of generality it is assumed thatGis connected and that ev- ery vertex ofGlies on a directedstpath.
A vector P= (p1; : : : ;pn) consisting of an st
path pi for each user i is a pure strategies profile. Let le(P) =Pi:e2piwibe the load of edgeeinP. The authors definethe cost i
Best Response Algorithms for Selfish Routing
B
87 pathpin the profilePto beip(P) = X e2p\pi de(le(P)) + X e2pXpi de(le(P) +wi):
The costi(P) of useriinPis justi
pi(P), i. e. the total delay along her path.
A pure strategies profilePis a Pure Nash Equilibrium (PNE) iff no user can reduce her total delay byunilaterally deviatingi. e. by selecting anotherstpath for her load, while all other users keep their paths.
Best Response
Letpibe the path of useriandPi =p1; : : : ;pibe the
pure strategies profile for users 1; : : : ;i. Then thebest re- sponseof useri+ 1 is a pathpi+1so that
pi+1=av gmin p2Pi 8 < : X e2p de le Pi+wi+1 9= ; : Flows and Common Best Response
A (feasible) flow on the setPofstpaths ofGis a func- tionf :P! <0so that X p2P fp= n X i=1 wi:
The single-commodity network congestion game
((wi)i2N;G;(de)e2E) has the Common Best Response
property if for every initial flowf (not necessarily feasible), all users have the same set of best responses with respect tof. That is, if a pathpis a best response with respect tof for some user, then for all usersjand all pathsp0
X e2p0 defe+wj X e2p defe+wj:
Furthermore, every segment of a best response pathpis a best response for routing the demand of any user between’s endpoints. It is allowed here that some users may already have contributed to the initial flowf. Layered and Series-Parallel Graphs
A directed (multi)graph G(V,E) with a distinguished sourcesand destinationt islayerediff all directedst
paths have exactly the same length and each vertex lies on some directedstpath.
A multigraph isseries-parallelwithterminals(s,t) if 1. it is a single edge (s,t) or
2. it is obtained from two series-parallel graphsG1;G2
with terminals (s1;t1) and (s2;t2) by connecting them
either inseriesor inparallel. In a series connection,t1
is identified withs2ands1becomessandt2becomest.
In a parallel connection,s1=s2=sandt1=t2=t.
Key Results
The Greedy Best Response Algorithm (GBR)
GBRconsiders the users one-by-one innon-increasingor- der of weight (i. e.w1w2 wn). Each user adopts
her best response strategy on the set of (already adopted in the net) best responses of previous users. No user can change her strategy in the future. Formally,GBRsucceeds
if the eventual profilePis a Pure Nash Equilibrium (PNE). The Characterization
In [3] it is shown:
Theorem 1 If G is an(st)series-parallel graph and the game((wi)i2N;G;(de)e2E)has the common best response property, thenGBRsucceeds.
Theorem 2 A weighted single-commodity network conges- tion game in a layered network with identical edges has the common best response property for any set of user weights.
Theorem 3 For any single-commodity network congestion game in series-parallel networks,GBRsucceeds if
1. The users are identical (if wi = 1for all i) and the edge- delays are arbitrary but non-decreasing or
2. The graph is layered and the edges are identical (for ar- bitrary user weights)
Theorem 4 If the network consists of bunches of parallel- links connected in series, then aPNEis obtained by applying GBRto each bunch.
Theorem 5
1. If the network is not series-parallel then there exist games whereGBRfails, even for 2 identical users and identical edges.
2. If the network does not have the common best response property (and is not a sequence of parallel links graphs connected in series) then there exist games whereGBR fails, even for 2-layered series-parallel graphs.
Examples of such games are provided in [3]. Applications
GBRhas a natural distributed implementation based on a leader election algorithm. Each player is now represented by a process. It is assumed that processes know the net- work and the edge latency functions. The existence of
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B
Bidimensionalitya message passing subsystem and an underlying synchro- nization mechanism (e. g. logical timestamps) is assumed, that allows a distributed protocol to proceed in logical rounds.
Initially all processes are active. In each round they run a leader election algorithm and determine the process of largest weight (among the active ones). This process routes its demand on its best response path, announces its strat- egy to all active processes, and becomes passive. Notice that each process can compute its best response locally. Open Problems
What is the class of networks where (identical) users can achieve aPNEby ak-round repetition of a best responses sequence? What happens to weighted users? In general, how the network topology affects best response sequences? Such open problems are a subject of current research. Cross References
General Equilibrium
Recommended Reading
1. Awerbuch, B., Azar, Y., Epstein, A.: The price of Routing Unsplit- table Flows. In: Proc. ACM Symposium on Theory of Comput- ing (STOC) 2005, pp. 57-66. ACM, New York (2005)
2. Duffin, R.J.: Topology of Series-Parallel Networks. J. Math. Anal. Appl.10, 303–318 (1965)
3. Fotakis, D., Kontogiannis, S., Spirakis, P.: Symmetry in Net- work Congestion Games: Pure Equilibria and Anarchy Cost. In: Proc. of the 3rd Workshop of Approximate and On-line Al- gorithms (WAOA 2005). Lecture Notes in Computer Science (LNCS), vol. 3879, pp. 161–175. Springer, Berlin Heidelberg (2006)
4. Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish Unsplittable Flows. J. Theor. Comput. Sci.348, 226–239 (2005)
5. Libman, L., Orda, A.: Atomic Resource Sharing in Noncoopera- tive Networks. Telecommun. Syst.17(4), 385-409 (2001)