Transcript Matching with Commitments

MATCHING WITH COMMITMENTS
Pushkar Tripathi
Georgia Institute of Technology
China Theory Week
2011
Joint work with Kevin Costello and Prasad Tetali
Objective : Maximize the number of goods exchanged
Model
Chen, Immorlica, Karlin, Mahdian, Rudra [ICALP 09]
e present or not
??
u
pe
Catch : If e is present then u and v are matched
Objective : Maximize the expected number of
vertices that get matched.
v
Approximation Factor
° =
min
І = G(V, p)
E[ ALG( І ) ]
E[ Max matching in G(V, p) ]
Compare against omniscient adversary who
knows the underlying graph
Greedy Matching

Try edges in arbitrary order.
Maximal matching in every instance.

½ approximation for every instance.

Greedy algorithm is a ½-approximate algorithm.
Bipartite Graphs
Simulate the RANKING algorithm [KVV 90]
For each arriving vertex,
query edges according to
ranking order
Ranking on
the vertices
Match each arriving vertex
to the highest available free
neighbor.
[KVV 90] : RANKING attains a factor of 1-1/e
Bipartite Graphs – 2 sided RANKING
Shuffle both sides and simulate the RANKING algorithm
Ranking on
The vertices
Match each arriving vertex to the
highest available free neighbor.
[MY, KMT 11] : 2-sided RANKING attains a factor of 0.69
General Graphs – Shuffle Algorithm
Aronson, Dyer, Frieze, Suen [STOC 95]
[ADFS 95] : SHUFFLE attains a factor of 0.50000025
Question : Can we beat the factor for ADFS by
using the stochastic information effectively ?
Results


0.573 factor algorithm running in O(n3) time.
No algorithm can achieve a factor better than
0.896.
Scanning in order of pe
0.99
1.0
0.99
Observation : pe is not a good measure of the
importance of an edge.
Define qe
qe = Pr[ e 2 Max. matching ]
0.99
1.0
0.99
0.0001
0.99
0.99
p - values
q - values
Claim : qe can be closely approximated by sampling
from the distribution without probing any edges
Easy Case : qe/pe is large
qe/pe ¸ ® > 0
Test e
1 – pe
pe
e not present
e present
<1-®
No damage
done.
>®
e 2 Max
Matching
e 2 Max
Matching
OPT reduces by at most 2
ALG increases by 1
OPT reduces by 1
ALG increases by 1
ALG
OPT
OPT
Bad case !!
Algorithm
// Easy Case.
While there is e such that qe/pe ¸ ®
Test edge e
Re-compute qe
// Begin Hard Case
……
Hard Case: qe/pe0 for all edges
pe = log(n)/n
qe = 1/n
qe/pe = 1/log(n)
Technical Lemma
p1
r1
p2
r2
p3
r3
p4
r4
p5
r5
p6
r6
Mild necessary
conditions
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
Proof of main lemma
Has no Solution !!
x1 ¸ x2 ¸ x3 …. ¸ xn ¸ 0
Identity
permutation
Sk = {1,2, … k} , 8 k 2 [n]
Multiply each equation by xi – xi+1
From previousContradiction
slide … =
=
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
Implication
p1
r1
¼ p2
r2
Stop when you find
the first edge
p3
r3
p4
r4
p5
r5
Conclusion : Each edge is chosen with probability at least qe
Back to bipartite graphs
p1
r1
¼ p2
r2
Qu =  qe
e 2 ±(u)
Every vertex tests
edges according to
a freshly chosen ¼
E[OPT] = uQu
u
Pr[ u is matched] ¸ 1- ∏(1-qe )
> 1 - e- qe
>  qe(1 – 1/e) = Qu(1 – 1/e)
p3
r3
p4
r4
p5
r5
Lemma : Sampling based algorithm also attains a factor
of 1-1/e for bipartite graphs
How about general graphs
Idea : Randomly partition the vertex set and restrict the
graph to a bipartite graph
Every vertex tests
edges according to
a freshly chosen ¼
u
u
½Qu
Pr[ u is matched] ¸ 1- ∏(1-qe )
> 1 - e- qe
>  qe(1 – 1/e)
= ½Qu(1 – 1/e)
Exploit qe/pe< ®
qe/pe< ® … think ® = 0.1
¯ ri
General graphs again….
v
Scale the
Requirements
by ¯
u
Pr[ u is matched] ¸ 1- ∏(1-¯qe )
> 1 - e- bqe
>  ¯ qe(1 – 1/e)
= ½ ¯ Qu(1 – 1/e)
Qv
Qv · ½
v
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
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.
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