MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011...
-
Upload
carol-mcclure -
Category
Documents
-
view
215 -
download
0
Transcript of MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011...
![Page 1: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/1.jpg)
MATCHING WITH COMMITMENTS
Joint work with Kevin Costello and Prasad TetaliChina Theory Week
2011
Pushkar TripathiGeorgia Institute of Technology
![Page 2: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/2.jpg)
Objective : Maximize the number of goods exchanged
![Page 3: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/3.jpg)
Model
pe
??
e present or not
Catch : If e is present then u and v are matched
Objective : Maximize the expected number of vertices that get matched.
uv
Chen, Immorlica, Karlin, Mahdian, Rudra [ICALP 09]
![Page 4: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/4.jpg)
Approximation Factor
° = min E[ ALG( І ) ]E[ Max matching in G(V, p) ]І = G(V, p)
Compare against omniscient adversary who knows the underlying graph
![Page 5: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/5.jpg)
Greedy Matching
Try edges in arbitrary order. Maximal matching in every instance.
½ approximation for every instance.
Greedy algorithm is a ½-approximate algorithm.
![Page 6: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/6.jpg)
Bipartite Graphs
Simulate the RANKING algorithm [KVV 90]
Match each arriving vertex to the highest available free neighbor.
Ranking on the vertices
[KVV 90] : RANKING attains a factor of 1-1/e
For each arriving vertex, query edges according to ranking order
![Page 7: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/7.jpg)
Bipartite Graphs – 2 sided RANKING
Shuffle both sides and simulate the RANKING algorithm
Match each arriving vertex to the highest available free neighbor.
Ranking on The vertices
[MY, KMT 11] : 2-sided RANKING attains a factor of 0.69
![Page 8: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/8.jpg)
General Graphs – Shuffle Algorithm
[ADFS 95] : SHUFFLE attains a factor of 0.50000025
Question : Can we beat the factor for ADFS by using the stochastic information effectively ?
Aronson, Dyer, Frieze, Suen [STOC 95]
![Page 9: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/9.jpg)
Results
0.573 factor algorithm running in O(n3) time.
No algorithm can achieve a factor better than 0.896.
![Page 10: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/10.jpg)
Scanning in order of pe
0.99
0.99
1.0
Observation : pe is not a good measure of the importance of an edge.
![Page 11: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/11.jpg)
Define qe
qe = Pr[ e 2 Max. matching ]
0.99
0.99
1.0
0.99
0.99
0.0001
q - valuesp - values
Claim : qe can be closely approximated by sampling from the distribution without probing any edges
![Page 12: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/12.jpg)
Easy Case : qe/pe is large
qe/pe ¸ ® > 0
OPT OPT
ALG
e not present
e present
Test e
No damage
done.
e 2 Max Matching
e 2 Max Matching
OPT reduces by at most 2ALG increases by 1
OPT reduces by 1ALG increases by 1
1 – pe pe
> ®< 1 - ®
Bad case !!
![Page 13: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/13.jpg)
Algorithm
// Easy Case.While there is e such that qe/pe ¸ ®
Test edge eRe-compute qe
// Begin Hard Case ……
![Page 14: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/14.jpg)
Hard Case: qe/pe0 for all edges
pe = log(n)/nqe = 1/nqe/pe = 1/log(n)
![Page 15: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/15.jpg)
![Page 16: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/16.jpg)
Technical Lemma
p1
r1
p2
r2
p3
r3
p4
r4
p5
r5
p6
r6
Lemma : There exists a distribution over Sn so that for ¼ drawn from this distribution : Pr[ Ai is the earliest occurring event in ¼ ] ¸ ri
Mild necessary conditions
![Page 17: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/17.jpg)
Proof of main lemma
x1 ¸ x2 ¸ x3 …. ¸ xn ¸ 0
Has no Solution !!
Identitypermutation
![Page 18: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/18.jpg)
Sk = {1,2, … k} , 8 k 2 [n]
Multiply each equation by xi – xi+1
=
=
From previous slide …Contradiction
![Page 19: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/19.jpg)
Comments
Distribution can be found by Linear Programming
Combinatorial algorithm that runs in quadratic time
r = q satisfies the necessary conditions
Conclusion : For any vertex we can sample its neighborhood so that each edge is chosen with probability at least qe
Sampled and found to exist
![Page 20: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/20.jpg)
Implication
p1
r1
p2
r2
p3
r3
p4
r4
p5
r5
¼
Stop when you find the first edge
Conclusion : Each edge is chosen with probability at least qe
![Page 21: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/21.jpg)
Back to bipartite graphsp1
r1
p2
r2
p3
r3
p4
r4
p5
r5
¼
Every vertex tests edges according toa freshly chosen ¼
Pr[ u is matched] ¸ 1- ∏(1-qe ) > 1 - e- qe > qe(1 – 1/e) = Qu(1 – 1/e)
u
Lemma : Sampling based algorithm also attains a factor of 1-1/e for bipartite graphs
Qu = qe
e 2 ±(u)
E[OPT] = uQu
![Page 22: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/22.jpg)
How about general graphs
Idea : Randomly partition the vertex set and restrict the graph to a bipartite graph
Every vertex tests edges according toa freshly chosen ¼
Pr[ u is matched] ¸ 1- ∏(1-qe ) > 1 - e- qe > qe(1 – 1/e) = ½Qu(1 – 1/e)
u
u
½Qu
![Page 23: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/23.jpg)
Exploit qe/pe< ®
qe/pe< ® … think ® = 0.1
¯ ri
![Page 24: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/24.jpg)
General graphs again….
Pr[ u is matched] ¸ 1- ∏(1-¯qe ) > 1 - e- b qe > ¯ qe(1 – 1/e) = ½ ¯ Qu(1 – 1/e)
u
v Qv
Qv · ½ v
Scale the Requirements by ¯
![Page 25: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/25.jpg)
Final Step – Concluding the hard case
Recurse on the remaining vertices Optimize ® and ¯ to balance the
performance of the algorithm for ‘easy’ and ‘hard’ case
Theorem : Our algorithm attains a factor of 0.573
![Page 26: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/26.jpg)
Optimizations
The sampling trick can be implemented combinatorially in quadratic time
Use approximate maximum matching while recalculating qe – Almost linear time
Delay re-computing qe after scanning every edge – Only log(m) phases of re-computation.
![Page 27: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/27.jpg)
Hard example
Optimal algorithm solves a stochastic DP with exponentially(in the number of edges) many states.
Solve this DP for G(4,p=0.64) E[Matching returned by optimal alg.] =
1.607 E[Max Matching in G(4,0.64) ] = 1.792
Theorem : No algorithm can achieve a factor better than 0.896
![Page 28: MATCHING WITH COMMITMENTS Joint work with Kevin Costello and Prasad Tetali China Theory Week 2011 Pushkar Tripathi Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649caf5503460f94972bc3/html5/thumbnails/28.jpg)