algorithms & lps for k-edge connected spanning subgraphs
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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