Tighter Cut-Based Bounds for k-pairs Communication Problems

1
Tighter Cut-BasedBounds for k-pairs
Communication Problems

• Nick Harvey
• Robert Kleinberg

2
Overview

• Definitions
• Sparsity and Meagerness Bounds
• Show these bounds very loose
• Define Informational Meagerness
• Based on Informational Dominance
• Show that it can be slightly loose

3
k-pairs Communication Problem
S(1)
S(2)
T(2)
T(1)
4
Concurrent Rate

• Source i desires communication rate di.
• Rate r is achievable if rate vector rd1, rd2,
  , rdk is achievable
• Rate region interval of R
• Def Network coding rate (or NCR) sup r
  r is achievable

5
k-pairs Communication Problem
S(1)
S(2)

• d1 d2 1ce 1 ?e?E
• Rate 1 achievable

T(2)
T(1)
6
Upper bounds on rate

• Classical Sparsity bound for multicommodity
  flows
• CT91 General bound for multi-commodity
  information networks
• B02 Application of CT91 to directed network
  coding instances equivalent to sparsity.
• KS03 Bound for undirected networks with
  arbitrarytwo-way channels
• HKL04 Meagerness
• SYC03, HKL05 LP bound
• KS05 Bound based on iterative d-separation

7
Vertex-Sparsity

• Def For U ? V,
• VS (G) minU?V VS (U)
• Claim NCR ? VS (G)

Capacity of edges crossing between U and
U Demand of commodities separated by U
VS (U)
8
Edge-Sparsity

• Def For A ? E,
• ES (G) minA?E ES (A)
• Claim Max-Flow ? ES (G)
• But Sometimes NCR gt ES (G)

Capacity of edges in A Demand of commodities
separated in G\A
ES (A)
9
NCR gt Edge-Sparsity
S(1)
S(2)
e
T(2)
T(1)

• Cut e separates S(1) and S(2)
• ? ES (e) 1/2
• But rate 1 achievable!

10
Meagerness

• Def For A ? E and P ? k,A isolates P if for
  all i,j ? P,S(i) and T(j) disconnected in G\A.
• M (G) minA?E M (A)
• Claim NCR ? M (G)

Capacity of edges in A Demand of commodities in P
M (A) minP isolated by A
11
Meagerness Vtx-Sparsity are weak
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Thm M (Gn) VS (Gn) ?(1),but NCR ? 1/n.

12
A Proof Tool

• Def Let A,B ? E. B is downstream of Aif B
  disconnected from sources in G\A.Notation A ?
  B.
• Claim If A ? B then H(A) ? H(A,B).
• Pf Because S ? A ? B form Markov chain.

13
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? gn,T(1),h1

14
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? gn,T(1),h1 ? S(1),f1,g1,h1

15
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? gn,T(1),h1 ? S(1),f1,g1,h1
• ? S(1),f1,T(2),h2

16
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? gn,T(1),h1 ? S(1),f1,g1,h1
• ? S(1),f1,T(2),h2 ? S(1),S(2),f2,g2,h2

17
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
h3
hn-1
h1
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? gn,T(1),h1 ? S(1),f1,g1,h1
• ? S(1),f1,T(2),h2 ? S(1),S(2),f2,g2,h2
• ? S(1),S(2),f2,T(3),h3

18
Lemma NCR ? 1/n
S(3)
S(2)
S(n)
S(n-1)
f2
fn-1
f3
S(1)
f1
g2
g3
g1
gn-1
gn
Gn
hn-1
h1
h3
h2
T(1)
T(n-1)
T(n)
T(3)
T(2)

• Proof
• gn ? ? S(1),S(2),,S(n)
• Thus 1 ? H(gn) ? H(S(1),,S(n)) nr
• So 1/n ? r

19
Towards a stronger bound

• Our focus cut-based bounds
• Given A ? E, we want to infer thatH(A) ? H(A,P)
  where P?S(1),,S(k)
• Meagerness uses Markovicity(sources in P) ? A ?
  (sinks in P)
• Markovicity sometimes not enough

20
Informational Dominance

• Def A dominates B if information in A determines
  information in Bin every network coding
  solution.Denoted A B.
• Trivially implies H(A) ? H(A,B)
• How to determine if A dominates B?
• HKL05 give combinatorial characterization and
  efficient algorithm to test if A dominates B.

21
Informational Meagerness

• Def For A ? E and P ? S(1),,S(k),A
  informationally isolates P ifA?P P.
• iM (A) minP for P informationally isolated by
  A
• iM (G) minA ? E iM (A)
• Claim NCR ? iM (G).

22
iMeagerness Example
s1
s2
t1
t2

• Obviously NCR 1.
• But no two edges disconnect t1 and t2 from both
  sources!

23
iMeagerness Example
s1
s2
Cut A
t1
t2

• After removing A, still a path from s2 to t1!

24
Informational Dominance Example
Cut A

• Our characterization shows A t1,t2
• H(A) ? H(t1,t2) and iM (G) 1

25
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26
A bad example Hn

• Thm iMeagerness gap of Hn is ?(log V)

27
Tn Binary tree of depth n Source S(i) ?i?Tn
28
Tn Binary tree of depth n Source S(i)
?i?Tn Sink T(i) ?i?Tn
29
Nodes q(i) and r(i) for every leaf i of Tn
30
Complete bip. graph between sources and qs
31
(r(a),t(b)) if b ancestor of a in Tn
32
(s(a),t(b)) if a and b cousins in Tn
33
All edges have capacity ? except (q(i),r(i))
Capacity 2-n
34
Demand of source at depth i is 2-i
Capacity 2-n
t(00)
t(01)
t(10)
t(11)
t(0)
t(1)
t(e)
35
Properties of Hn

• Lemma iM (Hn) ?(1)
• Lemma NCR lt 1/n
• Corollary iMeagerness gap is n?(log V)

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Properties of Hn

• Lemma iM (Hn) ?(1)
• Lemma NCR lt 1/n
• Corollary iMeagerness gap is nO(log V)

We will prove this
37
Proof Ingredients

• Entropy moneybags
• i.e., sets of RVs
• Entropy investments
• Buying sources and edges, putting into moneybag
• Loans may be necessary
• Profit
• Via Downstreamness or Info. Dominance
• Earn new sources or edges for moneybag
• Corporate mergers
• Via Submodularity
• New Investment Opportunities and Debt
  Consolidation
• Debt repayment

38
Submodularity of Entropy

• Claim Let A and B be sets of RVs.Then H(A)H(B)
  ? H(A?B)H(A?B)
• Pf Equivalent to I( X Y Z ) ? 0.

39
Lemma NCR lt 1/n

• Proof
• Two entropy moneybags
• F(a) S(b) b not an ancestor of a
• E(a) F(a) ? (q(b),r(b)) b is descendant of
  a

40
Entropy Investment
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
a
q(01)
q(10)
q(11)
q(00)

• Let a be a leaf of Tn
• Take a loan and buy E(a).

41
Earning Profit
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
a
q(01)
q(10)
q(11)
q(00)

• Claim E(a) ? T(a)
• Pf Cousin-edges not from ancestors.Vertex r(00)
  blocked by E(a).

t(00)
42
Earning Profit
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
a
q(01)
q(10)
q(11)
q(00)

• Claim E(a) ? T(a)
• ResultE(a) gives free upgrade to
  E(a)?S(a).Profit S(a).

t(00)
43
s(e)
s(0)
s(1)
aL
s(00)
s(01)
s(10)
s(11)
E(aL)?S(aL)
q(10)
q(01)
q(11)
q(00)
aR
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
E(aR)?S(aR)
q(01)
q(10)
q(11)
q(00)
44
Applying submodularity
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
(E(aL)?S(aL)) ? (E(aR)?S(aR))
q(10)
q(01)
q(11)
q(00)
s(e)
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
(E(aL)?S(aL)) ? (E(aR)?S(aR))
q(01)
q(10)
q(11)
q(00)
45
New Investment
s(e)
a
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
(E(aL)?S(aL)) ? (E(aR)?S(aR))
q(10)
q(01)
q(11)
q(00)

• Union term has more edges
• ? Can use downstreamnessor informational
  dominance again!
• (E(aL)?S(aL)) ? (E(aR)?S(aR)) E(a)

46
Debt Consolidation

• Intersection term has only sources
• ? Cannot earn new profit.
• Used for later debt repayment
• (E(aL)?S(aL)) ? (E(aR)?S(aR)) F(a)

s(e)
a
s(0)
s(1)
s(00)
s(01)
s(10)
s(11)
(E(aL)?S(aL)) ? (E(aR)?S(aR))
q(01)
q(10)
q(11)
q(00)
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What have we shown?

• Let aL,aR be sibling leaves a is their parent.
• H(E(aL)) H(E(aR)) ? H(E(a)) H(F(a))
• Iterate and sum over all nodes in tree
• where r is the root.
• Note E(v) F(v) ? (q(v),r(v)) when v is a leaf

48
Debt Repayment

• Claim
• Pf Simple counting argument.

?
49
Finishing up
? Rate lt 1/n