Algorithms & LPs for k-Edge Connected Spanning Subgraphs

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Algorithms &amp; 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.

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### Algorithms & LPs fork-Edge ConnectedSpanning 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-ecsmultisubgraph problem (k-ECSM): can buy asmany copiesas you likeof any edge

3-edge-connected multisubgraph of G, |E|=9

G

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

u

v

S

w

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

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 hardinstance for2-VCSS

zero-cost edges to V(G)

Instance for 3-VCSSwith 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-2cover

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-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)

+

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]

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|2vars, 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]
• 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?

Small Extreme Examples

n=7, Δ=4

n=6, denom=2

n=8, denom=3

n=9, Δ=5

n=10, denom=Δ=5

n=9, denom=4

Previously Known Constructions

[BP]: minimum nonzero value of x* can be ~1/|V|

[C]: max degree can be ~|V|1/2