Transcript Hypergraphic LP Relaxations for the Steiner Tree Problem

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
3-edge-connected
G
of any edge
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.
vertex-connected
For metric costs, can split-off conversely, e.g.
2-ECSM
2-ECSS
2-VCSS
All of these are APX-hard [via {1,2}-TSP]
What’s Hard About Hardness?
1+ε hardness for 2-VCSS implies 1+ε hardness
for k-VCSS, for all k ≥ 2
G
G, a hard
instance for
2-VCSS
zero-cost
edges to V(G)
Instance for 3-VCSS
with same hardness
But this approach fails for k-ECSS, k-ECSM
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
0 × (k-1) 0 × (k-1)
0 × (k-1)
0 × (k-1)
1
0 × (k-1) 0 × (k-1)
1
1
1
… min |X| to cover T =
k-ECSS optimum.
Part 2: Linear Programs
From Hardness to
Approximability
Conjecture [P.]
For some constant C, there is aan LP-relative
(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
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
S
∑e in δ(S) xe ≥ k
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
(integrality gap)
ALG
+
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
OPEN connected
[B-J Y 01] subdigraphs
Ǝk, every k-edge connected hypergraph
FALSE
[B-J Thypergraphs
03]
contains 2 disjoint
connected
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]
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=9, Δ=5
n=7, Δ=4
n=8, denom=3
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