Polylogarithmic Inapproximability of Radio Broadcast

1
PolylogarithmicInapproximability ofRadio
Broadcast

• Guy Kortsarz
• Joint work with Michael Elkin

2
Radio Broadcast

• Undirected graph, v ? V wants to broadcast
• A vertex receives the message if and only if
  exactly one its neighbors transmits

3
The radio broadcast problem

• Given a graph G,v
• Find a minimum number of rounds schedule.
• Let opt denote the optimum number of rounds

4
Example

• In the following graph the optimum is 3

R2
R1
R3
R3
Figure 1 opt 3
5
History

• Chlamtac and Weinstein, 87
• O(R? log2n) upper bound
• Bar-Yehuda, Goldreich and Itai, Kowalski and
  Pelc, 04
• O(R? log n log2n) upper bound
• Alon, Bar-Noy, Lineal and Peleg, 89
• R?(log2n) lower bound
• Gaber and Mansour, 95
• O(Rlog5n) upper bound
• Elkin and Kortsarz, 04
• RO(R1/2 log2n)O(Rlog4n) upper bound
• Elkin and Kortsarz, 04
• RO(R1/2 log n log3n)O(Rlog3n)
• for planar graphs

6
Approximation status
Table1 The summary of previous and our results
Ref Type Bound
EK02 M.L.B. log n
KP04 M.U.B. log n, R ? log n
EK04 B O(R log4n)
KP04 A.U.B. R O(log2n), R?? log n
This paper A.L.B. R o(log2n)
7
Min-Rep

• Input G(A,B,E)
• Given A partition A ?Ai, B ?Bi

8
Goal

• Choose overall few (representative) from X ? A ?
  B so that X is minimum, and
• All superedges are covered

B3
Figure 3 An exact solution
9
The MIN-REP hardness result

• In its full generality, due to Ran Raz
• Yes instance is mapped to an exact cover
• No instance every choice of complete cover needs
  average of
• representatives per Ai, Bi

10
The star property

• The hardness result holds even under the
  assumption of the star property

11
Set-Cover

• Input B(V1, V2, H)
• S ? V1 covers V2 if N(S) V2
• Goal Minimum size V2 cover

12
The Lund and Yanakakis L.B
B
Figure 6 SET-COVER
13
The Lund and Yanakakis L.B

• Yes instance An exact MIN-REP cover gives and
  exact cover
• No instance In a no instance, every cover is
  large
• log( M(A,B) ) gap

14
The reduction
B
15
A 3 rounds schedule for a YES instance
16
Witnesses for NO instance
Figure 9 Type 1 and Type 2 witnesses
17

• Choose all deleted vertices as Type 1 witnesses
• From every remaining round choose 2 witnesses.
  Type 2 witnesses
• If of rounds is O (log n), then of witnesses
  is O (log n)
• If v is not connected to all Type 1 witnesses,
  but is connected to all Type 2 witnesses, v
  doesnt get the message
• Pr 1/pol for that
• Use union bound over all schedules

18
Open problems

• Prove O (R log2n) upper bound
• If R O (log n) can we do better than log2n
  approximation?
• Prove R O (log2n)(?)
• or opt O (log2n)