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Algorithms & LPs fork-Edge Connected
Spanning Subgraphs
Dave Pritchard University of Waterloo
CMU Theory Lunch, Dec 2 ‘09
k-Edge Connected Graph
k edge-disjoint paths between every u, v
at least k edges leave S, for all ≠ S V∅ ⊊
(k-1) edge failures still leaves G connected
S
|δ(S)| ≥ k
k-ECSS & k-ECSM Optimization Problems
k-edge connected spanning subgraph problem:given an initial graph (possibly 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|=9
G
Overview of TalkAlgorithms/Complexity Linear Programs
Approximationalgorithms
Hardnessconstructions
Parsimonious Property
Alg.design
Vertex connectivitySubset k-ECSM
Intricate extreme
point solutions
TSP
Motivating Questions
What is the best possible approximation ratio (assuming P≠NP) for these problems?
What qualities of these various problems make them computationally easy or hard?
Can we learn some new useful broad techniques from the study of these problems?
Approximation: State of the Art
Unit Costs Arbitrary Costs
Lower bound
Upper bound
Lower bound
Upper bound
k-ECSS1+ε/k [GGTW]
~1+0.5/k [CT, GG]
1+ε/k[GGTW]
2[KV, J]
k-ECSM?1+ε, k=2
~1+1.9/k [GGTW,GG]
?1+ε, k=2
~3/2 [GB]
(Worst-case ratio from optimal)
1+O(1/k)?
1+ε [P.]
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
Sv
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 APX-hard, i.e. no 1+ε approx [BBHKPSU]
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, a hardinstance for
2-VCSS Instance for 3-VCSSwith same hardness
G
zero-cost edges to V(G)
k-ECSS is APX-hard (1/2)
We reduce MinTreeCoverByPaths 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
k-ECSS is APX-hard (2/2)
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
k-ECSS problem = min |X| to cover T.
0 x (k-1) 0 x (k-1)
0 x (k-1)0 x (k-1) 0 x (k-1) 0 x (k-1)
1
1 1 1
Part 2:Complexity ∩ Linear Programs
From Hardness to Approximability
Conjecture [P.]For some constant C, there is a(1+C/k)-approximation algorithm for k-ECSM.
Holds for C=1, k ≤ 2.
Next: definition of LP-relative;
similar theorems known to be true;
motivating consequence.
an LP-relative
LP-Relative (1/2)
Term LP-relative hides a specific reference to a particular “undirected” linear programming relaxation of the k-ECSM problem:
Introduce variables xe ≥ 0 for all edges e of G.
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-Relative (2/2)
k-ECSM corresponds to integral LP solutions, but LP also has fractional solutions
So LP-OPT ≤ OPT (of k-ECSM)
α-approx algorithm: ALG ≤ α k-OPT⋅
Definition: an algorithm is LP-relative α-apx ifALG ≤ α LP-OPT⋅
+ALGOPTLP-OPT
(integrality gap)
Similar True Theorems
Width W of an integer linear program is the max ratio of RHS entry to LHS coefficient in the same row. (In case of k-ECSM IP it is k)
Conj: “ 1+O(1/W) LP-rel approx for k-ECSM”∃
1+O(1/W) LP-rel holds, and is tight, for
sparse integer programs
multicommodity flow/covering in treesLP structure for k-ECSM ≈ multiflow in tree
Background:(Per-vertex) Network Design
In input, each vertex v has requirement rv ∈ Z
Objective: find a min-cost subgraph s.t. for all vertices u, v, there are at least min{ru, rv} edge-disjoint paths connecting u and v
Has a similar undirected LP relaxation:x(δ(S)) ≥ min{ru, rv} if S separates u from v
[GB] showed LP has parsimonious property: without loss of generality, x(δ({v})) = rv for all v
0.5
0.5 1.5
Consequence of Conjecture
Subset k-ECSM: rv {0,k} for all v∈
vertices are required (rv= k) or optional (rv= 0)
By parsimonious property, Subset k-ECSM has the same LP as k-ECSM on required subset
Consequence of parsimony: LP-relative α-approx algorithm for k-ECSM impliesa same quality approx for Subset k-ECSM
conj.
A Combinatorial Approach?
Is the following true for some constant C?
“For every A, B > 0,every (A + B + C)-edge-connected graph contains a disjoint A-ECSM and B-ECSM?”
Part 3:LPs & Extreme Point
Structure
LPs & Extreme Point Properties (Part 3)
Compare k-ECSM LP and Held-Karp TSP LP
Introduce standard structural properties
Show how this gives the elegant algorithm of [GGTW] for k-ECSS
We undertake goal of finding an object as unstructured as possible: [P.] extreme points on n vertices with maximum ∃
degree n/2 and minimum value 1/Fibonacci(n/2)
k-ECSM LP by any other name
k-ECSM (using parsimony):xe ≥ 0, x(δ(S)) ≥ k, x(δ({v})) = k
Held-Karp relaxation of TSP (“outer” form): xe ≥ 0, x(δ(S)) ≥ 2, x(δ({v})) = 2
Therefore these LPs (for all k) are the same up to uniform scaling i.e., x feasible for first iff 2x/k feasible for second
Structural Property [CFN]
Held-Karp LP is large (2|V|-1 constraints, |V|∼ 2 variables) but:
every extreme point / basic / vertex solution x has at most 2|V|-3 nonzero coordinates
only 2|V|-3 constraints are needed to uniquely define this x, and we can pick a well-structured such set (laminar family)
Note: some optimal solution is basic
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
What Is Known about HK?
[BP]: minimum nonzero value of x* can be ~1/|V|
[C]: max degreecan be ~|V|1/2
What Is New?
• Edge values of the form Fibi/Fib|V|/2 and 1 - Fibi/Fib|V|/2
• Maximum degree |V|/2
How Was It Found?
Computational methods plus some cleverness can enumerate all extreme points on a small number of vertices We got up to 10; Boyd & coauthors have data
available online up to 12
Look for most complex extreme points: Big maximum degree, big denominator
Try to find a pattern & prove it
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
Laminar Set-Family
Any S, T inhave S T,⊂T S, or⊂S,T disjoint
Maximal: cannot add any new sets and retain laminarity
Proof that this is indeed a family of extreme points
Need to show x* is feasible, extreme
First, show x*(δ(S))=2 holds for a maximal laminar system L*
Argue x* is unique such solution (long part)
Suppose x*(δ(S))<2 for some S Use uncrossing to show that we can find another
set S’ with x*(δ(S’))<2 and (S’ ∪ L*) laminar Contradicts maximality of L*, we are done
Could It Get Worse?
Determinant bound shows denominator of extreme point is at most ~|V||V|
Size of laminar family can be used to show max degree is at most n-3
This construction does not attain maximal denominator on 12 vertices
Review
We found a hardness construction fork-edge-connected spanning subgraph
No good hardness known for k-edge-connected spanning multisubgraph
LP-relative 1+O(1/k) algorithm for k-ECSM would give one for subset k-ECSM
Extremely extreme extreme points
Thesis Plug
Investigated hypergraphic LP relaxations of Steiner tree problem
Showed equivalences, structure, gap bounds
[joint with D. Chakrabarty & J. Könemann]
Thanks for Attending!
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