Edge Covering problems with budget constrains

1. Edge Covering problems with budget constrains By R. Gandhi and G. Kortsarz Presented by: Alantha Newman

2. First problem, unweighted case • Input: A graph G(V,E) and a number k • Required: A subset U of Vof size kthat is touched by the minimum number of edges. • Touching edges of U: An edge (u,v) sothat either uUor uU • We denote this number by t(U)

3. What is the properties of a good solution? Let us check the case of adregular graphs. The question is if the edges are internal or external S

4. What is the properties of a good solution In this example most of the edges in S stay inside. Which means that t(S) is close to kd/2 U

5. What is the properties of a good solution But S can behave badly. Namely most edges go to V-U In this case t(S) close to dk. U

6. A trivial ratio of 2 • Let OPT, |OPT|=k, be the best solution • Let U be the k least degrees vertices, thus deg(OPT)≥ deg(U) • Clearly t(OPT)≥deg(OPT)/2 • Therefore: • t(U)≤deg(U)≤deg(OPT) • ≤2t(OPT)

7. We tried to improve the 2 but there is a problem • The following we became aware of only recently. • Let G be a d-regular graph. Consider only “small” S and: = e(S,V-S)/deg(S) • The without restriction: the Sparsest Cut problem and admits a sqrt{log n}by AroraRao and Vazirani

8. The case of small S • What happens if we bound the size of S? • The question of if there is a small set S with bad expansion may be harder. • Let 0≤≤1/2 • And allow only sets with at most n vertices. What is the worst expansion?

9. The Small Set Expansion Conjecture • Let G be a d-regular graph.Let≤0.5. • Let =Min |S|≤ n e(S,V-S)/deg(S) • The SSEC: It is hard to tell between the following two cases, for small:≥ 1- and ≤ • Due to Raghavendra and Steurer.

10. Breaking the ratio of 2 is equivalent to disproving SSEC • For a given , k= n. If =e(S,V-S)/deg(S) is roughly one most edges are outside S. For our problem the value of the solution about kd

11. The other case • The good case is S with small expansion If =e(S,V-S)/deg(S) close to 0 almost all edges are internal to S. • This means the value for our problem may be very close to kd/2.Good for SSEC good for us.

12. The approximation of 2 is optimal(?) • An approximation better than 2 means that the Small Set Expansion Conjecture fails! • This problem reduces to the unique game conjecture by Khot. • Seems hugely hard but easier than the famed Unique Game conjecture. • In the words of the movie “Marathon Man:” Is it safe? (To assume its hard?)

13. Is the Small Graph Expasion conjecture set reliable? • Opinions vary. I think: VERY RELIABLE. • We tried to disprove the SSGEand failed • The first problem with ratio 2 so that breaking the ratio 2 disproves the conjecture is the At least size k Densest subgraph. I think unpublished (U. Feige). • I suspect that in order to give good ratio for the SSEC you need a good algorithm for the Dense k-subgraph

14. Our second main problem • Say that we are given a graph G(V,E) and a number M. Find the maximum number of vertices that are touched by at most M edges. • Again for this problem a ratio 2 is simple. • To the best of our knowledge this problem was first studied for approximation by Goldshmidt and Hochbaum • This problem is motivated by application in loading of semi-conductor components to be assembled into products

15. The weighted case • Vertices have weights. • Largest weight under M touching edges: Find a set U of maximum weight so that the number of edges touching U is at most M. • Minimum edges under cost at least k: Find a set U of cost at least k and minimize the number of edges touching U.

16. Our results • Minimum edges under cost at least k admits a polynomial time 2 ratio. Improves 2(1+) by Carnes and Shmoys. • Maximum weight under at most M edges: admits a polynomial time algorithm with ratio 2. Improves ratio 3 by Goldschmidt and Hochbaum.

17. Lower bounds • Goldschmidt and Hochbaumshow that these two problems are NPC. • Under the SSEC we show that our approximation is optimal for both problems. • We show: Both problems even in the unweighted case admits no 2- approximation for any constant >0 unless the SSEC fails.

18. Our results • The natural LP for Weight at least k and at mostMedges in the unweighted case, has integrality gap 2. Not a surprise given the SSEC. • However showing integrality gap 2 is quite non trivial. Uses the probabilistic method.

19. Comparing the two problems • We show that a ratio  for Minimum edges cost at leastk implies a ratio of for Maximum weight at mostMedges. It seems that the reverse does not hold.

20. Further results • We show that the density version of Minimum edges for cost at least k can be solved by flow (only LP solution was known). • Given a selected already set S the goal is to add a set U and minimize (e(U)+e(U,S))/deg(S)

21. Some ideas of how to give ratio 2 for Maximum cost at mostM edges • We use Dynamic Programing. • We guess the number P of edges between the optimum set OPTand V-OPT. • We guess the sum of degree of OPTwhom may be 2M. A serious technical problem: we are only able to compute A[n, P, M].

22. The reason for that • This is the only way, it seems, to assure feasibility. • Indeed if deg(U)≤M then t(U)≤M. • The question is do we loose a lot by bounding the sum of degrees by M while the sum may be 2M? • One more detail: we need to guess the highest cost vertex in OPT and add it the our solution.

23. If deg(U)≤M, how much cost we loose? • Let OPT= A {x}  B so that deg(A+x) isthe first to be above M • Thus A isa feasible solution for M. • Clearly B too is a feasible solution for Mbecause deg(A+x)>Mand the total at most 2M • One of A or B has ½ the weight. The fact that we guess the highest cost vertex in OPT compensate for x. • Thus ratio 2.

24. Thank you for your attention • Any Questions?