Universal Approximations in Network Design

1. Universal Approximations in Network Design Rajmohan Rajaraman Northeastern University Based on joint work with Chinmoy Dutta, Lujun Jia, Guolong Lin, Jaikumar Radhakrishnan, Ravi Sundaram, Emanuele Viola

2. 1,2,3,6? {1,2,3,6}: 20+15+15+30=90 {2,3,4,5}: 10+10+20+40=80 A day in the life of a courier 2,3,4,5? 1 8 2 7 4 3 6 5

3. 1,2,3,6? {1,2,3,6}: 20+15+15+30=90 20+40+15+20=95 {2,3,4,5}: 10+10+20+40=80 40+30+10+35=115 A day in…a lazy courier 2,3,4,5? Consider tour 1,2,3,4,5,6,7,8 1 8 2 7 4 3 6 5 Stretch ≥ 115/80

4. Is there a single tour that is universally good? Given any subset of the cities, the restricted subtour (preserving the order) is a good approximation to the best tour for that subset Is there a smart lazy courier? 1 8 2 7 4 3 6 5

5. Universal TSP • Given a metric space (V,d), design a tour T over V that minimizes • Tsis the sub-tour of T induced by S • Cost(X) is length of tour X • OPT(S) is cost of an optimal tour for S

6. Universal Steiner tree (UST) Candidate spanning tree T Set S OPT(S) = 6 Cost of induced Subtree TS = 16 Stretch ≥ 16/6

7. Universal Steiner tree (UST) • Given: A weighted graph G over set V of vertices, and a root vertex • Goal: Determine a spanning tree T of G that minimizes • Tsis the subtree of T induced by S and root • OPT(S) is cost of optimal tree connecting S to root • Cost(X) is the sum of weights of edges in X • Graphical UST: T only has edges from G • Metric UST: Work with metric completion of G • Graphical UST at least as hard as metric UST Stretch of T

8. Universal approximations framework • Universal version of optimization problem has two additional notions • Sub-instance relation ≤ • Restriction function R: Takes a solution S for instance I, a sub-instance I’ of I, and returns a solution R(S,I,I’) for I’ • Goal: For given instance I, determine a solution T for I that minimizes

9. Universal set cover • Given set V of elements, collection C of subsets of V, determine f: V  C such that • For all x in V, f(x) contains x, and • The following stretch is minimized

10. Motivation • Optimization under uncertainty • Universal solutions are robust against adversarial inputs • Aggregation tree in a sensor network • Data is being generated at several sensors, and aggregation queries arrive at a sink • Setting up aggregation trees dynamically as queries and data change may be expensive • Universal Steiner trees provide good approximations for all query and update patterns • Universal solutions are differentially private [Bhalgat-Chakrabarty-Khanna 11]

11. Universal approximation results

12. The roadmap • Landscape around universal approximations • Universal Steiner trees • Bounded locally consistent partitions • Metric UST • Graphical UST • Lower bound • Concluding remarks and open problems

13. The “universal” landscape • O(log n)-stretch universal TSP for the Euclidean plane [Platzman-Bartholdi 89] • Simultaneous approximations for single-sink buy-at-bulk • Given a graph, demands to be routed to a sink, cost for each edge, route demands to minimize total cost • A single tree is a simultaneous O(1)-approximation for all concave cost functions [Goel-Estrin 03,…,Goel-Post 10] • Tree decompositions • [Fakcharoenphol-Rao-Talwar 03] yields metric tree whose expected stretch for each set is O(log n) • O(log n loglog n) using [Elkin-Emek-Spielman-Teng 06, Abraham-Bartal-Neiman 09] distribution over spanning trees • The cut-based decompositions of [Räcke 02,08] also aim for a distribution over trees or tree with prob. embedding

14. The “universal” landscape • Oblivious routing and network design • Given graph, source-sink pairs, and per-edge routing cost, determine routes that are oblivious to demand pairs and cost function • O(log2n)-approximation for sub-additive cost functions • [Räcke 02, Harrelson-Hildrum-Rao 03, Gupta-Hajiaghayi-Räcke 06] • A priori approximations [Schalekamp-Shmoys 08] • For TSP, set of vertices visited drawn from a probability distribution • Set covering with eyes closed • Determine a single mapping of elements to sets to minimize expected cost of covering random element subset • [Grandoni-Gupta-Leonardi-Miettinen-Sankowski-Singh 08]

15. Universal Steiner tree (UST) • Given: A weighted graph G over set V of vertices, and a root vertex • Goal: Determine a spanning tree T of G that minimizes • Tsis the subtree of T induced by S and root • OPT(S) is cost of an optimal tree connecting S to root • Cost(X) is the sum of weights of edges in X Stretch of T

16. What does a good UST look like? Candidate spanning tree T

17. What does a good UST look like? • At each distance level, T provides a clustering of G • Given tree T, adversary identifies set S such that • S is “well-separated” in T • S is “close” in G • To avoid this, UST should cluster nodes so that • Each node’s neighborhood does not intersect too many clusters • Otherwise, adversary will select several nodes from this neighborhood lying in different clusters

18. Bounded locally consistent partitions • A partition of the metric space with the following properties: • Diameter of every cluster in partition is at most αR • Every R-ball intersects β clusters • Every metric space has an (O(log n),O(log n),R)-partition for every R • Sparse partitions [Awerbuch-Peleg 90], [Peleg 00] β = 4

19. Hierarchical partitions • A collection of partitions {Pi} with the following properties: • Partition: Pi is an (α,β,Ri)-partition • Hierarchy: Pi is a refinement of Pi+1 • Root padding: Cluster in Pi containing root contains ball of radius Riaround root • Every metric space has a hierarchical (O(logn), O(logn),O(logn))- partition

20. A metric UST algorithm [JLNRS 05] • Compute a hierarchical (O(log n),O(log n),O(log n))-partition • For each level i, from lowest to highest: • For each level i cluster: • Select leader from leaders of its constituent level i-1 clusters • connect level i leader to level i-1 leaders • Root is always leader of its clusters

21. Proof sketch for stretch • To prove: For every set S, Cost(TS) is at most polylog(n) times OPT(S) • For a level j, cost in UST is O(njlogj+1n): • njis the number of level-j ancestors of nodes in S • Main Lemma: • If Pj is a maximal set of nodes in S pair-wise separated by logj-1n • Then nj = O(|Pj| log n) • Cost(OPT(S)) is Ω(|Pj|logj-1n) • Cost at level j in UST is thus O(log3n)Cost(OPT(S))

22. Bounding the cost at a level Proof sketch of Main Lemma: • Any node’s ancestor at level j is within O(logjn) cost of node • Therefore, O(logjn)-ball around the ancestors of Pj at level j covers all nj ancestors of S at level j • By partitioning scheme, it follows that nj is O(|Pj|log n) Pj is maximal set of nodes in S pair-wise separated by logj-1n nj is the number of nodes at level j of induced tree We have nj = O(|Pj| log n)

23. Improved bounds for special metrics • For doubling metrics, the UST algorithm achieves a stretch of O(log n) • Hierarchical (O(1),O(1),O(1))-partition • Doubling metrics include Euclidean metrics as well as growth-restricted metrics

24. An O(log2n)-stretch metric UST • Gupta-Hajiaghayi-Räcke 06 • α-padding: A node v is α-padded in a hierarchical decomposition if • At level i, the ball of radius α2i around v is fully contained within its cluster at level i • Theorem: For any v, in any tree drawn from the [FRT 03] distribution, probability that v is Ω(1/logn)-padded is at least 3/4

25. An O(log2n)-stretch metric UST • Simple metric UST construction: • Sample O(log n) trees from the FRT distribution • For each vertex v select a tree where v is Ω(1/logn)-padded • In each tree, build the sub-tree induced by the root and vertices that selected the tree (using metric completion) • Return the union of the O(log n) sub-trees computed above • O(log2n) stretch

26. Challenges for Graphical UST • Bounded locally consistent partition: • Partition G into clusters of strong diameter at most αR • Each R-ball intersects at most β clusters • How small can α and β be? • Open: Is (polylog(n), polylog(n),1)-partitioning achievable? • Lemma (Necessity): If σ-stretch achievable for graphical UST, then (σ,σ2,R)-partition exists for all R.

27. Challenges for Graphical UST • Hierarchical partition: • Unlike in the metric case, cannot simply elect leaders and connect directly • Connecting lower level partitions arbitrarily may introduce huge blowup in costs • In the [GHR 06] approach: • Can replace the O(log n) FRT trees by the spanning trees drawn from [EEST 05] distribution • Not clear how to combine paths drawn from these trees into a single spanning tree

28. Graphical UST construction • Construct (2Õ(√logn), 2Õ(√logn),R)-partitions • (O(1),O(1),R) for doubling graphs • Convert to a hierarchical partitioning: • (2Õ(√logn),2Õ(√logn),2Õ(√logn)) for general graphs • (O(1), O(1), O(log2n)) for doubling graphs • Build UST from hierarchical partition: • Connect lower-level trees using shortest paths • Invoke properties of partitioning to bound stretch • for general graphs and 2Õ(√logn)for doubling graphs • [Dutta-Radhakrishnan-R-Sundaram-Viola 11]

29. Lower bound for UST • Every algorithm for on-line Steiner trees over n nodes has a competitive ratio of • (log n) for metrics [Imase-Waxman 91] • (log n/loglog n) for Euclidean metrics [Alon-Azar 92] • Any UST for an n-node metric space with stretch s(n) can be transformed into an on-line algorithm with competitive ratio of s(n) • Consequence: Every UST has a stretch of (log n) for n-node metrics, (log n/loglog n) for Euclidean metrics

30. Complexity of universal problems • For a given terminal set S: • Finding OPT(S) is NP-hard • Poly-time O(1)-approximations known (Minimum spanning tree,…,[]) • For a candidate UST, finding the worst-case set is NP-hard • Finding whether there exists a UST with stretch at most σ is coNP-hard • Universal problems are “”-optimization problems • The -quantification is over an exponential-sized domain • Lies in ∑2 • Open: is it ∑2-hard?

31. Open problems • Close the gaps for UTSP and metric UST • Euclidean UTSP: Ω(log1/6n) vs O(log n) • UTSP: Ω(log n) vs O(log2 n) • Metric UST: Ω(log n) vs O(log2n) • Is there a polylog(n)-stretch graphical UST? • Strong diameter partitions: • Can we partition any graph into components of strong diameter polylog(n) such that each vertex has neighbors in polylog(n) components? • [Peleg 00] • Universal approximations for other optimization problems