Polylogarithmic Inapproximability of Radio Broadcast Guy Kortsarz Joint work with: Michael Elkin

Radio Broadcast  Undirected graph, v  V wants to broadcast  A vertex receives the message if and only if exactly one its neighbors transmits

The radio broadcast problem  Given: a graph G,v  Find: a minimum number of rounds schedule. Let opt denote the optimum number of rounds

Example In the following graph the optimum is 3 R1 R3 R2 R3 Figure 1: opt = 3

History  Chlamtac and Weinstein, 87: O(R  log 2 n) upper bound O(R  log 2 n) upper bound  Bar-Yehuda, Goldreich and Itai, Kowalski and Pelc, 04: O(R  log n + log 2 n) upper bound O(R  log n + log 2 n) upper bound  Alon, Bar-Noy, Lineal and Peleg, 89: R+  (log 2 n) lower bound R+  (log 2 n) lower bound  Gaber and Mansour, 95: O(R+log 5 n) upper bound O(R+log 5 n) upper bound  Elkin and Kortsarz, 04: R+O(R 1/2 log 2 n)=O(R+log 4 n) upper bound R+O(R 1/2 log 2 n)=O(R+log 4 n) upper bound  Elkin and Kortsarz, 04: R+O(R 1/2 log n + log 3 n)=O(R+log 3 n) R+O(R 1/2 log n + log 3 n)=O(R+log 3 n) for planar graphs for planar graphs

Approximation status Approximation status RefTypeBound EK02M.L.B. log n KP04M.U.B. log n, R  log n EK04B O(R + log 4 n) KP04A.U.B. R + O(log 2 n), R  log n This paper A.L.B. R + o(log 2 n) Table1: The summary of previous and our results

Min-Rep 1. Input: G(A,B,E) 2. Given: A partition A =  A i, B =  B i A1A1 A2A2 A3A3 A4A4 B1B1 B2B2 B3B3 B4B4 aa’a’ Figure 2: A MIN-REP instance b

Goal 1. Choose overall few (representative) from X  A  B so that |X| is minimum, and: 2. All “superedges” are covered B3B3 a A1A1 A2A2 A3A3 A4A4 B1B1 B2B2 B4B4 b Figure 3: An “exact” solution

The MIN-REP hardness result In its full generality, due to Ran Raz 1. Yes instance is mapped to an “exact cover” 2. No instance: every choice of complete cover needs average of representatives per A i, B i representatives per A i, B i

The star property The hardness result holds even under the assumption of the star property: A B Figure 4: A  B induces a collection of stars

Set-Cover 1. Input: B(V 1, V 2, H) 2. S  V 1 covers V 2 if N(S) = V 2 3. Goal: Minimum size V 2 – cover Figure 5: SET-COVER

The Lund and Yanakakis L.B B Figure 6: SET-COVER A M(A,B) a b b’b’ b ’’

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

The reduction A M(A,B) a b b’b’ b ’’ B s Figure 7: a, b, b ’, b ’’ are connected to a random half of the complementary half

A 3 rounds schedule for a YES instance A a b B s R1 R2 R3 M(A,B) Figure 8: First s transmits, then  A  S transmits, and then  B  S transmits

Witnesses for NO instance Figure 9: Type 1 and Type 2 witnesses QQ Q Q Y Y YZ Z Z X X X X P P P

 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 doesn’t get the message  Pr = 1/pol for that  Use union bound over all schedules

Open problems 1. Prove O (R + log 2 n) upper bound 2. If R = O (log n) can we do better than log 2 n approximation? 3. Prove R + O (log 2 n)(?) or opt + O (log 2 n) or opt + O (log 2 n)