Algorithms & LPs for k-Edge Connected Spanning Subgraphs David Pritchard Zinal Workshop, Jan 20 ‘09

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

k-ECSS & k-ECSM Optimization Problems k-edge connected spanning subgraph problem (k-ECSS): given an initial graph (maybe 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 G 3-edge-connected multisubgraph of G, |E|=9

Approximation Algorithms k-ECSS and k-ECSM are NP-hard for k ≥ 2 For k-ECS/M or any NP-hard minimization problem, one notion of a good heuristic is a polytime “α-approximation algorithm”: always produces a feasible solution (e.g., k-edge connected subgraph) whose output cost is at most α times optimal Also call α the approximation ratio/factor

Approximability Any problem has either no constant-factor approximation algorithm Ǝ constant α ≥ 1: α-approx alg exists, but no (α-ε)-approx exists for any ε>0 [α=1: “in P”] Ǝ constant α ≥ 1: (α+ε)-approx alg exists for any ε>0, but no α-approx [α=1: “apx scheme”]

Question What is the approximability of the k-edge-connected spanning subgraph and k-edge-connected spanning multisubgraph problems? It will depend on k Exact approximability not known but we’ll determine rough dependence on k Also look at important unit-cost special case

History of k-ECSS 2-ECSS is NP-hard, has a 2-apx alg [FJ 80s] & is APX-hard (has no approximation scheme if P ≠ NP) even for unit costs [F 97] k-ECSS has a 2-apx alg [KV 92] unit cost k-ECSS: has a (1+2/k)-apx [CT96] & Ǝ tiny ε>0, for all k>2, no (1+ε/k)-apx [G+05] Gets easier to approximate as k increases! How do k-ECSS, k-ECSM depend on k?

Main Course We prove k-ECSS with general costs does not have approximation ratio tending to 1 as k tends to infinity We conjecture that k-ECSM with general costs does have approximation ratio tending to 1 as k tends to infinity

Part 1: Approximation Hardness

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 v S 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 of these are APX-hard [via {1,2}-TSP] vertex-connected 2-ECSM 2-ECSS 2-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 G, a hard instance for 2-VCSS zero-cost edges to V(G) Instance for 3-VCSS with same hardness

Hardness of k-ECSS (slide 1/2) Ǝ ε>0, ∀ k≥2, no 1+ε-apx if P ≠ NP Reduce APX-hard TreeCoverByPaths 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

Hardness of k-ECSS (slide 2/2) Ǝ ε>0, ∀ k≥2, no 1+ε-apx if P ≠ NP 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 … min |X| to cover T = k-ECSS optimum. 0 × (k-1) 0 × (k-1) 0 × (k-1) 0 × (k-1) 0 × (k-1) 0 × (k-1) 1 1 1 1

Part 2: Linear Programs

From Hardness to Approximability Conjecture [P.] For some constant C, there is a (1+C/k)-approximation algorithm for k-ECSM. For k ≤ 2 it is known to hold with C=1. Next: definition of LP-relative, motivation an LP-relative

LP Relaxation Introduce variables xe ≥ 0 for all edges e of G. LP relaxation of k-ECSM problem: min ∑ xecost(e) s.t. x(δ(S)) ≥ k for all ∅ ≠ S ⊊ V 0.4 ∑e in δ(S) xe ≥ k S 1.2 1.4

LP-Relative a new name for an old concept The LP is a linear relaxation of k-ECSM so LP-OPT ≤ OPT (optimal k-ECSM cost) Definition: an LP-relative α-approximation algorithm has output cost ALG ≤ α⋅LP-OPT (Stronger than “α-approx” — ALG ≤ α·OPT) + LP-OPT OPT ALG (integrality gap)

Motivation Similar results for related problems are true LP-relative (1+C/k)-approx for k-ECSM would imply a (1+C/k)-approx for subset k-ECSM Subset k-ECSM: given a graph with some vertices required, find a graph with edge-connectivity ≥ k between all required vertices (k=1: Steiner tree) To show this, use parsimonious property [GB93] of the LP: there is an optimal fractional solution that doesn’t use non-required vertices

A Combinatorial Approach? “Ǝ constant C so that for every A, B > 0, every (A + B + C)-edge-connected graph contains a disjoint A-ECSM and B-ECSM” would imply the conjecture. Cousins: Ǝk, each k-strongly connected digraph has 2 disjoint strongly connected subdigraphs Ǝk, every k-edge connected hypergraph contains 2 disjoint connected hypergraphs OPEN [B-J Y 01] FALSE [B-J T 03]

More About the LP LP relaxation of k-ECSM equals (up to scaling) the Held-Karp TSP relaxation, and the undirected Steiner tree LP relaxation. LP and generalizations are well-studied Basic (“extreme”) solutions have few nonzeroes Extreme solution properties are often key to algorithmic results (e.g., [GGTW 05]) How “unstructured” can extreme points get?

Extremely Extreme Extreme Point Edge values of the form Fibi/Fib|V|/2 and 1 - Fibi/Fib|V|/2 (exponentially small in |V|) Maximum degree |V|/2

Structural LP Property [CFN] The LP has 2|V|-1 constraints, |V|2 vars, but Ǝ a basic solution which is optimal (all LPs); we can find it in polytime (reformulation); every basic solution has ≤ 2|V|-3 nonzero coordinates (laminar family of constraints).

1+O(1/k) Algorithm for Unit-Cost k-ECSM [GGTW] Solve LP to get a basic optimal solution x* 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

Open Questions Resolve the conjecture! What’s the minimum f(A, B) so that any f(A, B)-edge-connected (multi)graph contains disjoint A- and B-edge-connected subgraphs? Is there a constant k such that every k-strongly connected digraph has 2 disjoint strongly connected subdigraphs?

Thanks for Attending!

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

Previously Known Constructions [BP]: minimum nonzero value of x* can be ~1/|V| [C]: max degree can be ~|V|1/2