1. Algorithms & LPs for k-Edge Connected Spanning Subgraphs Dave Pritchard University of Waterloo CMU Theory Lunch, Dec 2 ‘09

2. k-Edge Connected Graph k edge-disjoint paths between every u, v at least k edges leave S, for all ∅ ≠ S ⊊ V (k-1) edge failures still leaves G connected S |δ(S)| ≥ k

3. k-ECSS & k-ECSM Optimization Problems k-edge connected spanning subgraph problem: given an initial graph (possibly with edge costs), find k-edge connected subgraph including all vertices, w/ |E| (or cost) minimal k-ecs multisubgraph problem (k-ECSM): can buy as many copies as you like of any edge 3-edge-connected multisubgraph of G, |E|=9 G

4. Overview of Talk Algorithms/ComplexityLinear Programs Approximation algorithms Hardness constructions Parsimonious Property Alg. design Vertex connectivity Subset k-ECSM Intricate extreme point solutions TSP

5. Motivating Questions What is the best possible approximation ratio (assuming P≠NP) for these problems? What qualities of these various problems make them computationally easy or hard? Can we learn some new useful broad techniques from the study of these problems?

6. Approximation: State of the Art Unit CostsArbitrary Costs Lower bound Upper bound Lower bound Upper bound k-ECSS 1+ε/k [GGTW] ~1+0.5/k [CT, GG] 1+ε/k [GGTW] 2 [KV, J] k-ECSM ? 1+ε, k=2 ~1+1.9/k [GGTW,GG] ? 1+ε, k=2 ~3/2 [GB] (Worst-case ratio from optimal) 1+O(1/k)? 1+ ε [P.]

7. An Initial Observation For the k-ESCM (multisubgraph) problem, we may assume edge costs are metric, i.e. cost(uv) ≤ cost(uw) + cost(wv) since replacing uv with uw, wv maintains k-EC u S v w

8. What’s Hard About Hardness? A 2-VCSS is a 2-ECSS is a 2-ECSM. For metric costs, can split-off conversely, e.g. All APX-hard, i.e. no 1+ε approx [BBHKPSU] 2-ECSM2-ECSS2-VCSS

9. What’s Hard About Hardness? 1+ε hardness for 2-VCSS implies 1+ε hardness for k-VCSS, for all k ≥ 2 But this approach fails for k-ECSS, k-ECSM G, a hard instance for 2-VCSS Instance for 3-VCSS with same hardness G zero-cost edges to V(G)

10. k-ECSS is APX-hard (1/2) We reduce MinTreeCoverByPaths to k-ECSS Input: a tree T, collection X of paths in T A subcollection Y of X is a cover if the union of {E(p) | p in Y} equals E(T) Goal: min-size subcollection of X that is a cover size-2 cover

11. k-ECSS is APX-hard (2/2) Replace each edge e of T by k-1 zero-cost parallel edges; replace each path p in X by a unit-cost edge connecting endpoints of p k-ECSS problem = min |X| to cover T. 0 x (k-1) 1 111

12. Part 2: Complexity ∩ Linear Programs

13. From Hardness to Approximability Conjecture [P.] For some constant C, there is a (1+C/k)-approximation algorithm for k-ECSM. Holds for C=1, k ≤ 2. Next: definition of LP-relative; similar theorems known to be true; motivating consequence. an LP-relative

14. LP-Relative (1/2) Term LP-relative hides a specific reference to a particular “undirected” linear programming relaxation of the k-ECSM problem: Introduce variables x e ≥ 0 for all edges e of G. Min ∑ x e cost(e) s.t. x(δ(S)) ≥ k for all ∅ ≠ S ⊊ V S ∑ e in δ(S) x e ≥ k 0.4 1.2 1.4

15. LP-Relative (2/2) k-ECSM corresponds to integral LP solutions, but LP also has fractional solutions So LP-OPT ≤ OPT (of k-ECSM) α-approx algorithm: ALG ≤ α ⋅ k-OPT Definition: an algorithm is LP-relative α-apx if ALG ≤ α ⋅ LP-OPT + ALGOPTLP-OPT (integrality gap)

16. Similar True Theorems Width W of an integer linear program is the max ratio of RHS entry to LHS coefficient in the same row. (In case of k-ECSM IP it is k) Conj: “ ∃ 1+O(1/W) LP-rel approx for k-ECSM” 1+O(1/W) LP-rel holds, and is tight, for sparse integer programs multicommodity flow/covering in trees LP structure for k-ECSM ≈ multiflow in tree

17. Background: (Per-vertex) Network Design In input, each vertex v has requirement r v ∈ Z Objective: find a min-cost subgraph s.t. for all vertices u, v, there are at least min{r u, r v } edge-disjoint paths connecting u and v Has a similar undirected LP relaxation: x(δ(S)) ≥ min{r u, r v } if S separates u from v [GB] showed LP has parsimonious property: without loss of generality, x(δ({v})) = r v for all v 0.5 1.5

18. Consequence of Conjecture Subset k-ECSM: r v ∈ {0,k} for all v vertices are required (r v = k) or optional (r v = 0) By parsimonious property, Subset k-ECSM has the same LP as k-ECSM on required subset Consequence of parsimony: LP-relative α- approx algorithm for k-ECSM implies a same quality approx for Subset k-ECSM conj.

19. A Combinatorial Approach? Is the following true for some constant C? “For every A, B > 0, every (A + B + C)-edge-connected graph contains a disjoint A-ECSM and B-ECSM?”

20. Part 3: LPs & Extreme Point Structure

21. LPs & Extreme Point Properties (Part 3) Compare k-ECSM LP and Held-Karp TSP LP Introduce standard structural properties Show how this gives the elegant algorithm of [GGTW] for k-ECSS We undertake goal of finding an object as unstructured as possible: [P.] ∃ extreme points on n vertices with maximum degree n/2 and minimum value 1/Fibonacci(n/2)

22. k-ECSM LP by any other name k-ECSM (using parsimony): x e ≥ 0, x(δ(S)) ≥ k, x(δ({v})) = k Held-Karp relaxation of TSP (“outer” form): x e ≥ 0, x(δ(S)) ≥ 2, x(δ({v})) = 2 Therefore these LPs (for all k) are the same up to uniform scaling i.e., x feasible for first iff 2x/k feasible for second

23. Structural Property [CFN] Held-Karp LP is large (2 |V| -1 constraints, ∼ |V| 2 variables) but: every extreme point / basic / vertex solution x has at most 2|V|-3 nonzero coordinates only 2|V|-3 constraints are needed to uniquely define this x, and we can pick a well- structured such set (laminar family) Note: some optimal solution is basic

24. 1+O(1/k) Algorithm for Unit-Cost k-ECSM [GGTW] 1. Solve LP to get a basic optimal solution x* 2. Round every value in x* up to the next highest integer and return the corresponding multigraph (k=4)

25. Analysis Optimal k-ECSM has degree k or more at each vertex, hence at least k|V|/2 edges The fractional LP solution x* has value (fractional edge count) k|V|/2 There are at most 2|V|-3 nonzero coordinates Rounding up increases cost by at most 2|V|-3 ALG/OPT ≤ (k|V|/2 + 2|V|-3)/(k|V|/2) < 1 + 4/k

26. What Is Known about HK? [BP]: minimum nonzero value of x* can be ~1/|V| [C]: max degree can be ~|V| 1/2

27. What Is New? Edge values of the form Fib i /Fib |V|/2 and 1 - Fib i /Fib |V|/2 Maximum degree |V|/2

28. How Was It Found? Computational methods plus some cleverness can enumerate all extreme points on a small number of vertices We got up to 10; Boyd & coauthors have data available online up to 12 Look for most complex extreme points: Big maximum degree, big denominator Try to find a pattern & prove it

29. Small Extreme Examples n=6, denom=2 n=7, Δ=4 n=8, denom=3 n=9, Δ=5 n=9, denom=4 n=10, denom=Δ=5

30. Laminar Set-Family Any S, T in have S ⊂ T, T ⊂ S, or S,T disjoint Maximal: cannot add any new sets and retain laminarity

31. Proof that this is indeed a family of extreme points Need to show x* is feasible, extreme First, show x*(δ(S))=2 holds for a maximal laminar system L* Argue x* is unique such solution (long part) Suppose x*(δ(S))<2 for some S Use uncrossing to show that we can find another set S’ with x*(δ(S’))<2 and (S’ ∪ L*) laminar Contradicts maximality of L*, we are done

32. Could It Get Worse? Determinant bound shows denominator of extreme point is at most ~|V| |V| Size of laminar family can be used to show max degree is at most n-3 This construction does not attain maximal denominator on 12 vertices

33. Review We found a hardness construction for k-edge-connected spanning subgraph No good hardness known for k-edge- connected spanning multisubgraph LP-relative 1+O(1/k) algorithm for k-ECSM would give one for subset k-ECSM Extremely extreme extreme points

34. Thesis Plug Investigated hypergraphic LP relaxations of Steiner tree problem Showed equivalences, structure, gap bounds [joint with D. Chakrabarty & J. Könemann]

35. Thanks for Attending!