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

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

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

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

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?

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.]

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

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

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)

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

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

Part 2: Complexity ∩ Linear Programs

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

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

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)

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

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

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.

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?”

Part 3: LPs & Extreme Point Structure

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)

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

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

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)

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

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

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

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

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

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

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

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

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

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

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