Covering Crossing Biset-Families

by Digraphs

Zeev NutovThe Open University of Israel

2 years ago…

The problem can be casted as covering a crossing set-family by edges

Constant ratio for edge-costs; O(log n) ratio for node-costs.

GK:

GK:

ZN:

This is the first thing I check..

Any approximation for

problem Π? ZN:

How did you know it will be crossing??

Well …

Talk Outline

Intersecting and crossing set-families; Applications: Edge-connectivity problems. Ratio 2 for covering crossing set-families.

Intersecting and crossing biset-families; Applications: Node-connectivity problems. Logarithmic and almost constant ratios for covering crossing biset-families.

Intersecting and crossing set-families

Intersecting and crossing set-families

Two sets X,Y on a groundset V: intersect if X∩Y ≠ . cross if X∩Y ≠ and X⋃Y ≠ V.

A set-family F is an intersecting/crossing set-family if X∩Y, X⋃YF for any intersecting/crossing X,YF.

A directed edge covers a set S if it goes from S to V-S.

The Set-Family Edge-Cover problem

S V-S

The set-family F may not be given explicitly. We require

that certain queries on F are answered in polynomial time;

Given s,tV, return the inclusion-minimal/maximal set in F that contains s but not t.

Set-Family Edge-Cover Given: A graph (V,E) with edge-costs, set-family F on V.Find: Min-cost edge-cover JE of F.

Examples

Rooted Edge-Connectivity AugmentationGiven: ℓ-edge-connected to r graph G, edge-set E with

costs. Find: Min-cost JE so that G+J is (ℓ+1)-edge-connected to

r.

Both problems are particular cases of Set-Family Edge-Cover: Rooted Edge-Connectivity Augmentation – intersecting F.Global Edge-Connectivity Augmentation – crossing F.

Global Edge-Connectivity AugmentationGiven: ℓ-connected graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-edge-connected.

Examples-cont.

Intersecting families - arise in rooted connectivity problems.

Let G be a directed graph that is ℓ-edge-connected to r (namely, has ℓ edge-disjoint paths from r to every vV-r). Then the set-family F = {SV-r : dG(S)=ℓ} is intersecting.Crossing families - arise in global connectivity

problems.Let G be a directed graph that is ℓ-edge-connected(namely, has ℓ edge-disjoint paths between any two

nodes).Then the set family F = {SV : dG(S)=ℓ} is crossing.

dG(S) = the number of edges in G leaving S.

Approximability of Set-Family Edge-Cover

Set-Family Edge-Cover with intersecting F can be solved in polynomial time via a primal-dual algorithm [Frank

1999].

1. Choose rV; let F + = {SF : rV-S}, F

− = {SF : rS}. % The family F

+ and the family {V-S : SF −} of F − -complements are both intersecting.2. Return J= J

+ ⋃ J −;

J + is an optimal edge-cover of F

+ ; J

− is an optimal “reverse edge-cover” of F −-

complements.

What about Set-Family Edge-Cover with crossing F ?Can be decomposed into two problems of covering anintersecting set-family; thus admits a 2-approximation.

Intersecting and crossing biset-families

Bisets

A biset is an ordered pair of sets S=(SI,SO) with SI SO; SI is the inner part and SO is the outer part of S.

Intersection and the union of bisets X,Y are defined by:

X ∩ Y=(XI ∩ YI , XO ∩ YO) X ⋃ Y=(XI ⋃ YI , XO ⋃ YO)

Intersecting and crossing biset-families

Two bisets X,Y on a groundset V: intersect if XI∩YI ≠ . cross if XI∩YI ≠ and XO ⋃ YO ≠ V.

Bifamily is a biset family that is: bijective: X=Y if XI=YI or if XO=YO ; monotone: XO YO if XI YI.

A biset-family F is an intersecting/crossing biset-family if X∩Y, X⋃YF for any intersecting/crossing X,YF.

A directed edge covers a biset S if it goes from SI to V-SO.

The Bifamily Edge-Cover problem

SI V-SO

dG(S) = number of edges in G covering S.

The bifamily F may not be given explicitly …

Bifamily Edge-Cover Given: A graph (V,E) with edge-costs, bifamily F on V.Find: Min-cost edge cover JE of F.

SO

γ(S) = |SO-SI|

Examples

Rooted Connectivity AugmentationGiven: ℓ-connected to r graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-connected to r.

Global Connectivity AugmentationGiven: ℓ-connected graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-connected.

By Menger’s Theorem, both problems are particular cases

of Bifamily Edge-Cover. Rooted Connectivity Augmentation – intersecting F.Global Connectivity Augmentation – crossing F.

Examples-cont.Intersecting bifamilies - rooted connectivity problems.Let G be a directed graph that is ℓ-connected to r (has ℓ node-disjoint dipaths from r to every vV-r). The following bifamily (“violated” bisets) is intersecting F = {(SI,SO): SISOV-r, γ(S)+dG(S) = ℓ}

Crossing families - arise in global connectivity problems.Let G be a directed graph that is ℓ-connected(has ℓ node-disjoint dipaths between any two nodes).Then the following bifamily (“violated” bisets) is

crossing F = {(SI,SO): SISOV, γ(S) = ℓ, dG(S)=0}

SISO

r

Approximability of Bifamily Edge-Cover

Bifamily Edge-Cover with intersecting F can be solved in polynomial time via a primal-dual algorithm [Frank

2009].

Can we get a better ratio?

What about Bifamily Edge-Cover with crossing F ?Can be decomposed into 2(ℓ+1)- problems of covering

anintersecting bifamily; thus admits a 2(ℓ+1)-

approximation,where ℓ= max {γ(S) : SF}.

Logarithmic approximation

Theorem 1: Bifamily Edge-Cover with crossing F admits a polynomial time algorithm that computes an F-cover J of cost c(J) = τ · O(log ν) = τ · O(log n).

e E

. 1

0

τ

F

e e

ee S

e

min c x

s.t x S

x e E

ν = number of F-cores (inclusion-minimal sets in {SI:S∈F })

τ = the optimal value of a natural LP-relaxation

Almost constant approximation

Theorem 2: Bifamily Edge-Cover with crossing ℓ-regular F admits a polynomial time algorithm that computes an

F-cover J of cost , .

n

c J O log min nn

Corollary: The problem of increasing the connectivity of a graph from ℓ to ℓ+1 at minimum cost admits a polynomial time algorithm that computes a solution

J of cost , .

n

c J O log min nn

A bifamily F is ℓ-regular if γ(S) = ℓ for all S∈F and γ(X∩Y) ≥ ℓ for any intersecting X,Y∈F.

Proof-Sketch of Theorem 1

Theorem 1: Bifamily Edge-Cover with crossing F admits a polynomial time algorithm that computes an F-cover J of cost c(J) = τ · O(log ν) = τ · O(log n).

For a partial solution J, let FJ be the “residual bifamily”of F w.r.t. J (consists of members of F uncovered by J).

The Main Lemma: Bifamily Edge-Cover admits a polynomial time algorithm

that

computes an edge set J so that: c(J) ≤ τ and ν(FJ) ≤ ν(F)/2.

Proof-Sketch of Theorem 2

Observation: It is sufficient to show such an algorithm for the bifamily S ={S∈F : |SI|≤ q} where q = (n-ℓ)/2. (To cover F we apply this algorithm twice: once on F and once on the “reverse” bifamily of F.)

Lemma: |SI| ≤ q for all S∈S and for any intersecting X,Y∈S:

- X∩Y∈S (S is “intersection-closed”). - X⋃Y∈S if |XI⋃YI| ≤ q. We call such a bifamily q-semi-intersecting.

Theorem 2: Bifamily Edge-Cover with crossing ℓ-regular F admits a polynomial time algorithm that computes an

F-cover J of cost , .

n

c J O log min nn

O(log (n-ℓ))-approximation

Lemma: The S-cores (inclusion-minimal sets in {SI:S∈S })

are pairwise disjoint, and there exists a polynomial time

algorithm that finds an edge set of cost ≤ τ so that every “new” core contains two “old” cores.

Observation: We also have c(J) ≤ τ · log2ν(S).

AnalysisAfter i iterations 2i ≤|C| ≤ (n-ℓ)/2 for every SJ-core C.The number of iterations ≤ log2 (n-ℓ)/2=O(log (n-ℓ)).

Algorithm J←, and repeatedly add to J an edge-set as in the

Lemma.

-approximation

Theorem 3: Bifamily Edge-Cover with q-semi-intersecting S admits a polynomial time algorithms that computes an edge set I of cost ≤ τ so that ν(SI) ≤ n/(q+1).

nO log

n

Algorithm 1. Find edge-set I as in Theorem 3 to reduce the

number of cores to n/(q+1).

2. Find edge-set J, c(J) ≤ τ · log2ν(SI) (the Observation).Analysis (recall that q = (n-ℓ)/2)

c(I) ≤ τc(J) ≤ τ · log2 n/(q+1) = τ · O(log (n/(n-ℓ)).

Proof of Theorem 3’

Theorem 3’: Set-Family Edge-Cover with q-semi-intersecting S admits a polynomial time algorithms that computes an edge set I of cost ≤ τ so that ν(SI) ≤ n/(q+1).

Notation: For a subfamily U S of pairwise disjoint sets let

S(U)={SS: SU for some UU }.

High-Level Idea: Find an optimal edge cover of “large” S(U).

Observation: The family S(U) is intersecting.

LP and Complementary Slackness

min min

(P) s.t. 1 (D) s.t.

0 0

Se eSe E

e S ee S e S

e S

yc x

x S y c e E

x e E y S

F

F

F

Primal C.S conditions: eI e is tight

Dual C.S. conditions: yS>0 dI(S)=1

Primal-Dual Algorithm

Phase 1: While I does not cover S do: Raise the dual variable of an SI-core C until some edge e in E\I covering C becomes

tight. U← U + C─ {sets of U contained in C} I ← I + e

EndWhile

Initialization: I←.

Phase 2: Apply Reverse-Delete like the family S(U) is the

one we want to cover.

Analysis

ν(SI) ≤ n/(q+1) because it can be proved that:

The members of U are pairwise disjoint.

Any UU intersects at most one SI-core. For any SI-core C the union BC of C

and the sets of U intersecting C is not in S(U). Thus |BC| ≥ q+1.

c(I) ≤ τ(S(U)), since I,y satisfy the C.S. conditions

(S(U) is an intersecting family).

Consequently: The sets BC are pairwise

disjoint. |BC|≥ q+1 for any SI-core C.Q.E.D.

U

C

Summary and Open Questions

Summary of Ratios for Bifamily Edge-Cover:

intersecting F polynomial crossing F O(log n)

crossing ℓ-regular F

Increasing connectivity from ℓ to ℓ+1: the same ratio.

(Previous ratio was O(log ℓ) [FL STOC 08])

Open Questions Can we obtain a constant ratio? Conjecture: No.

Can we please go to dinner?

, .

nO log min n

n

Thanks!

Questions?