Distributed Computing Group Locality and the Hardness of Distributed Approximation Thomas Moscibroda Joint work with: Fabian Kuhn, Roger Wattenhofer

2Thomas DANSS 2004 Locality... Communication in multi-hop networks is inherently local ! Issue of locality is crucial in distributed systems! Direct communication only between neighbors Obtaining information about distant nodes requires multi-hop communication Obtaining information from entire network requires plenty of communication!

3Thomas DANSS 2004 Locality... Many modern networks are large-scale and highly complex –Internet –Peer-to-Peer Networks –Wireless Sensor Networks Or even dynamic... –Wireless Ad Hoc Networks  No node has global information  Each node has information from its vicinity only (local information)  Yet, nodes have to come up with a global goal!

4Thomas DANSS 2004 Example: Global Goal – Local Information Clustering in Wireless Sensor Networks Choose Clusterheads such that –Every node is either a clusterhead or... –...has a clusterhead in its neighborhood. Idea: Clusterhead sense environment Non-clusterheads can go to sleep mode  Save energy! Goal: We want as few clusterheads as possible! (Minimum Dominating Set Problem)  Nodes have only local information  Nodes have to optimize a global goal Crucial for fast Algorithms!

5Thomas DANSS 2004 k-Neighborhood What does „local“ mean ? How far does „locality“ go? Neighborhood, 2-Hop Neighborhood,... or something else...??? In communication round: 12 3

6Thomas DANSS 2004 k-Neighborhood In k rounds of communication each node can gather information only from k-neighborhood! If message size is unbounded:  Entire information from k-neighborhood (IDs, topology, edge-weights,...) can be collected! If message size is bounded:  Only subset of this information can be gathered. Strongest model for lower bounds on locality!

7Thomas DANSS 2004 What can be computed locally? [Naor,Stockmeyer;1993] We want to establish a trade-off between amount of communication and quality of the global solution. TRADE-OFF LOCALITY Communication Rounds GLOBAL GOAL Approximation Upper Bounds:  Constant-Time Approximation Algorithms Lower Bounds:  Hardness of Distributed Approximation

8Thomas DANSS 2004 What can be computed locally? [Naor,Stockmeyer;1993] How well can global tasks be locally approximated? –Minimum Dominating Set (Choose minimum S µ V, s.t. each v 2 V is in S or has at least one neighbor in S) –Minimum Vertex Cover (Choose minimum S µ V, s.t. each e 2 E has one node in S) Both problems appear to be local in nature!

9Thomas DANSS 2004 What can be computed locally? [Naor,Stockmeyer;1993]  An answer to the previous question......helps in answering the following question about exact variants of the problems. How large must the locality be in order to compute a maximal independent set or maximal matching?  An answer to this question implies important time-lower bounds for distributed algorithms!

10Thomas DANSS 2004 Overview Introduction to Locality Related Work Vertex Cover Upper Bound Vertex Cover Lower Bound Conclusions & Open Problems

11Thomas DANSS 2004 Related Work On Locality Naor, Stockmeyer 1993: Which locally checkable labelings can be computed in constant time? [Naor,Stockmeyer;1993] O(log n) algorithm for maximal independent set [Luby;1986] O(log n) algorithm for maximal matching [Israeli, Itai;1986] 3-coloring in a ring in O(log*n) time [Cole, Vishkin;1986] O(log*n) was shown to be optimal by Linial [Linial;1992]  Only previous lower bound on locality!

12Thomas DANSS 2004 Related Work On Distributed Approximation Upper Bounds (examples...) –Minimum Dominating Set Problem [Jia, Rajaraman, Suel; 2001] [Kuhn, Wattenhofer; 2003],... –Minimum Edge-Coloring [Panconesi, Srinivasan; 1997],... –General covering and packing problems [Bartal, Byers, Raz; 1997][Kuhn, Moscibroda, Wattenhofer; submitted] –Facility Location Lower Bounds –Results based on Linials  (log*n) lower bound –Recently, strong lower bound on the distributed approximability for the MST. [Elkin; 2004] For general graphs, we drastically improve this result.

13Thomas DANSS 2004 Overview Introduction to Locality Related Work Vertex Cover Upper Bound Vertex Cover Lower Bound Conclusions & Open Problems

14Thomas DANSS 2004 Minimum Vertex Cover We consider the most basic coordination Problem  Minimum Vertex Cover (MVC)  Choose as few nodes as possible to cover all edges We give an approximation algorithm with...  O(k) communication rounds  O(  1/k ) approximation  O(log n) bits message size General idea: Consider the integer linear program of MVC. Distributed Primal-Dual Approach

15Thomas DANSS 2004 Minimum Vertex Cover (Fractional) MVC is captured by the following linear program Its dual is the fractional maximum matching (MM) problem

16Thomas DANSS 2004 Each node stores a value x i Each edge stores a value y j –In a real network: edge is simulated by incident node! Idea: –Compute a feasible solution for MVC –While doing so: Distribute the dual values y j among incident, uncovered edges, hence –We show that  This yields an O(  1/k ) approximation! Distributed MVC Algorithm y1y1 y2y2 y3y3 xixi =1 =1/3 y4y4

17Thomas DANSS 2004 Distributed MVC Algorithm Number of incident, uncovered edges Maximum  i in neighborhood If relative number of uncovered edges is high  join VC If sum of y j in neighborhood is ¸ 1:  Pick node and distribute y j proportionally! It can be shown that:  i ·  (l+1)/k

18Thomas DANSS 2004 Analysis Lemma: At the end of the algorithm, for all nodes v i 2 V: Y i · 3+  1/k Idea: Bound the sum of the incident dual variables for each node i. Proof: Let  i denote the i th iteration of the loop Case 1: Consider a node which does not join the vertex cover! 1)Until  0,, Y i is smaller than 1 2)In  0,  i must be 0 3)All neighbors must have joined VC before  0  Y i · 1

19Thomas DANSS 2004 Before  l : Y i · 1 During  l : x i := 1  Y i · 2 neighbors v k may join VC, too  increase Y i All these nodes have  k ¸  i (1) l/(l+1) ¸  i l/(l+1) We get at most 1/  k from each of the  v uncovered neighbors!   v /  k ·  1/k Analysis Case 2: Consider a node that joins VC in line 5 of iteration  l. before  l : vivi during  l : 0.33 vkvk 0.62 Y i · 3+  1/k Additional nodes joining VC can increase Y i by at most 1 Case 3 is similar

20Thomas DANSS 2004 Summary VC-Algorithm Algorithm runs in O(k) rounds  Equivalent: Locality is O(k) hops! Message size is O(log n) bits  Approximation quality is  1/k +O(1) How many rounds are necessary for a O(1) or O(polylog  ) approximation? O(log  ) time  O(1) approximation O(log  /loglog  ) time  O(polylog  ) approximation Can we do better?

21Thomas DANSS 2004 Overview Introduction to Locality Related Work Vertex Cover Upper Bound Vertex Cover Lower Bound Conclusions & Open Problems

22Thomas DANSS 2004 Model Network graph = graph on which we compute VC Nodes have unique identifiers Message size and local computation are unbounded Strongest possible model for lower bounds  Lower bounds are consequence of locality alone

23Thomas DANSS 2004 Basic Idea S 0 and S 1 contain n 0 and n 0 /  nodes, resp. Optimal VC does not contain nodes of S 0 Basic Structure of our proof: 1.Construct graph such that nodes in S 0 and S 1 have same view 2.Algorithm has to take nodes of S 0 in order to cover edges between S 0 and S 1 3.VC ALG >> VC OPT because |S 0 | >> |S 1 | S 0 S 1

24Thomas DANSS 2004 One Round Lower Bound S 1 S 0

25Thomas DANSS 2004 One Round Lower Bound S 0 S 1

26Thomas DANSS 2004 S 0 S 1 Two Round Lower Bound

27Thomas DANSS 2004 S 0 S View of node in S 0 View of node in S 1 Two Round Lower Bound: Views

28Thomas DANSS 2004 The Cluster Tree I

29Thomas DANSS 2004 The Cluster Tree II Cluster Tree = a tree of clusters of nodes Recursively defined for k>0 Defines the structure of a graph G k Each link on the tree is a bipartite sub-graph of G k If girth of G k is at least 2k+1, nodes in S 0 and S 1 have the same view up to distance k

30Thomas DANSS 2004 Construction of G k How can we achieve high girth? G k is a bipartite graph (even level clusters / odd level clusters) For prime power q, D(r,q) is bipartite graph with 2q r nodes and girth at least r+5 [Lazebnik,Ustimenko; Explicit Construction of Graphs With an Arbitrary Large Girth and of Large Size; 1995] If  >k, G k can be constructed as sub-graph of D(2k-4,q) for q=  O(k)  G k has n=  O(k 2 ) nodes This is according to Intuition......because every node in S 0 must see a tree up to distance k.

31Thomas DANSS 2004 Bounding the Optimum The number of nodes decreases by factor at least  /k on each level. If  >2k, n < n 0 +2n 0 k/  All nodes V \ S 0 form a feasible vertex cover, hence |VC OPT | < 2n 0 k/  n 0: · n 0 k/  : · n 0 k 2 /  2 : · n 0 k 3 /  3 : geometric series

32Thomas DANSS 2004 Bounding any distributed algorithm Assume that the labeling (IDs) is chosen uniformly at random: Nodes v 0 in S 0 and v 1 in S 1 see –Same topology –Same probability distribution of labels –Both have same probability for being in VC (probability p) p is at least ½,  otherwise there is a probability that VC is not feasible!  Therefore: At least half of the nodes in S 0 join VC! For all algorithms, there is labeling with |VC ALG | ¸ n 0 /2 Randomized: |VC ALG | ¸ n 0 /2 by Yao’s minimax principle v0v0 v1v1

33Thomas DANSS 2004 Approximation Lower Bound We have |VC ALG | ¸ n 0 /2 and |VC OPT |  /(4k) n =  O(k 2 ),  =  k+1 In k communication rounds, no algorithm can approximate MVC better than  n c/k 2 /k) or  (  1/k /k)

34Thomas DANSS 2004 Time Lower Bound For constant/polylog approximation, we need Recall our vertex cover algorithm O(log  ) time  O(1) approximation O(log  /loglog  ) time  O(polylog  ) approximation Can we do better? Algorithm tight for polylog approximation and tight up to O(loglog  ) for constant approximation!

35Thomas DANSS 2004 Hardness of Approximation  Exact Problems Approximation Theory is very active area of research (see STOC, FOCS, SODA,...) Study of lower bounds on approximation  Hardness of Approximation! This has lead to new insight in complexity theory (PCP,...) Study of Approximability Better understanding of exact problems!

36Thomas DANSS 2004 Hardness of Approximation  Exact Problems Maximal matching (MM) is 2-approximation for MVC… (MM) is maximal independent set (MIS) on line graph, … Does the same hold in distributed computing? To some degree, it does....! Time lower bounds for MIS and Maximal Matching Compare with  log*n) lower bound on ring and O(log n) upper bound

37Thomas DANSS 2004 What about Dominating Sets ? For each VC instance, there is graph on which dominating set is the same  MVC bounds also hold for MDS Approximation lower bound can also be extended to maximum matching (more than just a reduction)

38Thomas DANSS 2004 Conclusions Locality is vital in distributed systems. Not much is known so far... In this talk, lower bounds on local computation  tight up to a factor of and  Vertex Cover, Dominating Set, Maximum Matching, MIS, Maximal Matching,... The hardness of distributed approximation is an interesting research topic

39Thomas DANSS 2004 Distributed Computing Group Questions? Comments?