Algorithms & LPs fork-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

S

|δ(S)| ≥ k

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 asmany copiesas you likeof any edge 3-edge-connected

multisubgraph of G, |E|=9G

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 andk-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

uS

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 of these are APX-hard [via {1,2}-TSP]2-ECSM 2-ECSS 2-VCSS

vertex-connected

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 hardinstance for

2-VCSS Instance for 3-VCSSwith same hardness

G

zero-cost edges to V(G)

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-2cover

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

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

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∅ ⊊

S ∑e in δ(S) xe ≥ k0.4

1.2

1.4

LP-Relativea 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)

+ALGOPTLP-OPT

(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, andthe 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 forUnit-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

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=6, denom=2 n=7, Δ=4 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 degreecan be ~|V|1/2