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!