Covering Crossing Biset -Families by Digraphs

1. Covering Crossing Biset-Families by Digraphs ZeevNutovThe Open University of Israel

2. 2 years ago… Any approximation for problem Π? GK: ZN: • The problem can be casted as covering a crossing set-family by edges  • Constant ratio for edge-costs; • O(log n) ratio for node-costs. How did you know it will be crossing?? GK: This is the first thing I check.. ZN: Well …

3. Talk Outline • Intersecting and crossing set-families; • Applications: Edge-connectivity problems. • Ratio 2 for covering crossing set-families. • Intersecting and crossing biset-families; • Applications: Node-connectivity problems. • Logarithmic and almost constant ratios for covering crossing biset-families.

4. Intersecting and crossing set-families

5. Intersecting and crossing set-families • Two sets X,Y on a groundset V: • intersect if X∩Y ≠. • cross if X∩Y ≠ and X⋃Y ≠ V. • A set-family F is an intersecting/crossing set-familyif X∩Y, X⋃YF for any intersecting/crossing X,YF.

6. The Set-Family Edge-Cover problem A directed edge covers a set S if it goes from S to V-S. S V-S Set-Family Edge-Cover Given: A graph (V,E) with edge-costs, set-family F on V. Find: Min-cost edge-cover JE of F. The set-family F may not be given explicitly. We require that certain queries on F are answered in polynomial time; Given s,tV, return the inclusion-minimal/maximal set in F that contains s but not t.

7. Examples Rooted Edge-Connectivity Augmentation Given: ℓ-edge-connected to r graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-edge-connected to r. Global Edge-Connectivity Augmentation Given: ℓ-connected graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-edge-connected. Both problems are particular cases of Set-Family Edge-Cover: Rooted Edge-Connectivity Augmentation – intersecting F. Global Edge-Connectivity Augmentation – crossing F.

8. Examples-cont. dG(S) = the number of edges in G leaving S. Intersecting families - arise in rooted connectivityproblems. Let G be a directed graph that is ℓ-edge-connected to r (namely, has ℓ edge-disjoint paths from r to every vV-r). Then the set-family F = {SV-r : dG(S)=ℓ} is intersecting. Crossing families - arise in global connectivity problems. Let G be a directed graph that is ℓ-edge-connected (namely, has ℓ edge-disjoint paths between any two nodes). Then the set family F = {SV : dG(S)=ℓ} is crossing.

9. Approximability of Set-Family Edge-Cover Set-Family Edge-Cover with intersectingF can be solved in polynomial time via a primal-dual algorithm [Frank 1999]. What about Set-Family Edge-Cover with crossingF ? Can be decomposed into two problems of covering an intersecting set-family; thus admits a 2-approximation. Choose rV; let F+ = {SF : rV-S}, F− = {SF : rS}. % The family F+ and the family {V-S : SF −} of F − -complements are both intersecting. Return J=J+ ⋃J−; J+ is an optimal edge-cover of F+ ; J− is an optimal “reverse edge-cover” of F−-complements.

10. Intersecting and crossing biset-families

11. Bisets A biset is an ordered pair of sets S=(SI,SO) with SI SO; SI is the inner part and SO is the outer part of S. Intersection and the union of bisets X,Y are defined by: X∩Y=(XI ∩YI , XO ∩YO) X⋃Y=(XI⋃YI , XO⋃YO)

12. Intersecting and crossing biset-families • Two bisets X,Y on a groundset V: • intersect if XI∩YI≠. • cross if XI∩YI≠ and XO ⋃YO≠ V. • A biset-family F is an intersecting/crossing biset-family if X∩Y, X⋃YF for any intersecting/crossing X,YF. • Bifamily is a biset family that is: • bijective: X=Y if XI=YI or if XO=YO ; • monotone: XO  YO if XI  YI.

13. The Bifamily Edge-Cover problem A directed edge covers a bisetS if it goes from SI to V-SO. dG(S) = number of edges in G covering S. γ(S) =|SO-SI| SI V-SO SO Bifamily Edge-Cover Given: A graph (V,E) with edge-costs, bifamily F on V. Find: Min-cost edge cover JE of F. The bifamilyF may not be given explicitly …

14. Examples Rooted Connectivity Augmentation Given: ℓ-connected to r graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-connected to r. Global Connectivity Augmentation Given: ℓ-connected graph G, edge-set E with costs. Find: Min-cost JE so that G+J is (ℓ+1)-connected. By Menger’s Theorem, both problems are particular cases of Bifamily Edge-Cover. Rooted Connectivity Augmentation – intersecting F. Global Connectivity Augmentation – crossing F.

15. Examples-cont. Intersecting bifamilies - rooted connectivity problems. Let G be a directed graph that is ℓ-connected to r (has ℓ node-disjoint dipaths from r to every vV-r). The following bifamily (“violated” bisets) is intersecting F = {(SI,SO): SISOV-r, γ(S)+dG(S) = ℓ} r SI SO Crossing families - arise in global connectivity problems. Let G be a directed graph that is ℓ-connected (has ℓnode-disjoint dipaths between any two nodes). Then the following bifamily (“violated” bisets) is crossing F = {(SI,SO): SISOV, γ(S) = ℓ, dG(S)=0}

16. Approximability of Bifamily Edge-Cover Bifamily Edge-Cover with intersectingF can be solved in polynomial time via a primal-dual algorithm [Frank 2009]. What about Bifamily Edge-Cover with crossingF ? Can be decomposed into 2(ℓ+1)- problems of covering an intersecting bifamily; thus admits a 2(ℓ+1)-approximation, where ℓ= max {γ(S) : SF}. Can we get a better ratio?

17. Logarithmic approximation Theorem 1: Bifamily Edge-Cover with crossing F admits a polynomial time algorithm that computes an F-cover J of cost c(J) = τ · O(log ν) =τ · O(log n). ν = number of F-cores (inclusion-minimal sets in {SI:S∈F}) τ= the optimal value of a natural LP-relaxation

18. Almost constant approximation A bifamilyF is ℓ-regular if γ(S) =ℓ for all S∈F and γ(X∩Y) ≥ℓ for any intersecting X,Y∈F. Theorem 2: Bifamily Edge-Cover with crossing ℓ-regular F admits a polynomial time algorithm that computes an F-cover J of cost Corollary: The problem of increasing the connectivity of a graph from ℓ to ℓ+1 at minimum cost admits a polynomial time algorithm that computes a solution J of cost

19. Proof-Sketch of Theorem 1 Theorem 1: Bifamily Edge-Cover with crossing F admits a polynomial time algorithm that computes an F-cover J of cost c(J) = τ · O(log ν) =τ · O(log n). For a partial solution J, let FJ be the “residual bifamily” of Fw.r.t. J (consists of members of F uncovered by J). The Main Lemma: Bifamily Edge-Cover admits a polynomial time algorithm that computes an edge set J so that: c(J) ≤ τ and ν(FJ) ≤ ν(F)/2.

20. Proof-Sketch of Theorem 2 Theorem 2: Bifamily Edge-Cover with crossing ℓ-regular F admits a polynomial time algorithm that computes an F-cover J of cost Observation: It is sufficient to show such an algorithm for the bifamilyS={S∈F : |SI|≤ q} where q = (n-ℓ)/2. (To cover F we apply this algorithm twice: once on F and once on the “reverse” bifamily of F.) Lemma: |SI| ≤ q for all S∈S and for any intersecting X,Y∈S: - X∩Y∈S (S is “intersection-closed”). - X⋃Y∈S if |XI⋃YI| ≤ q. We call such a bifamily q-semi-intersecting.

21. O(log (n-ℓ))-approximation Lemma: The S-cores (inclusion-minimal sets in {SI:S∈S}) are pairwise disjoint, and there exists a polynomial time algorithm that finds an edge set of cost ≤ τso that every “new” core contains two “old” cores. Algorithm J←, and repeatedly add to J an edge-set as in the Lemma. Analysis After i iterations 2i ≤|C| ≤ (n-ℓ)/2for every SJ-core C. The number of iterations ≤ log2 (n-ℓ)/2=O(log (n-ℓ)). Observation: We also have c(J) ≤ τ · log2ν(S).

22. -approximation Theorem 3: Bifamily Edge-Cover with q-semi-intersecting S admits a polynomial time algorithms that computes an edge set I of cost ≤ τso that ν(SI) ≤ n/(q+1). Algorithm Find edge-set I as in Theorem 3 to reduce the number of cores to n/(q+1). 2. Find edge-set J, c(J) ≤ τ · log2ν(SI) (the Observation). Analysis (recall that q = (n-ℓ)/2) c(I) ≤τ c(J) ≤τ ·log2 n/(q+1) = τ ·O(log (n/(n-ℓ)).

23. Proof of Theorem 3’ Theorem 3’: Set-Family Edge-Cover with q-semi-intersecting S admits a polynomial time algorithms that computes an edge set I of cost ≤ τso that ν(SI) ≤ n/(q+1). Notation: For a subfamily U S of pairwise disjoint sets let • S(U)={SS: SU for some UU }. Observation: The family S(U) is intersecting. High-Level Idea: Find an optimal edge cover of “large” S(U).

24. LP and Complementary Slackness Primal C.S conditions: eI  e is tight Dual C.S. conditions: yS>0  dI(S)=1

25. Primal-Dual Algorithm Initialization: I←. Phase 1: While I does not cover S do: • Raise the dual variable of an SI-core C until some edge e in E\I covering C becomes tight. • U←U + C─ {sets of U contained in C} I ← I + e EndWhile Phase 2: Apply Reverse-Delete like the family S(U) is the one we want to cover.

26. Analysis c(I) ≤ τ(S(U)), since I,y satisfy the C.S. conditions (S(U) is an intersecting family). ν(SI) ≤ n/(q+1)because it can be proved that: • The members of U are pairwise disjoint. • Any UUintersects at most one SI-core. • For any SI-core C the union BC of C and the sets of U intersecting C is not in S(U). Thus |BC| ≥ q+1. U Consequently: • The sets BC are pairwise disjoint. • |BC|≥ q+1 for any SI-core C. Q.E.D. C

28. Summary and Open Questions SummaryofRatios for Bifamily Edge-Cover: intersecting F polynomial crossing F O(log n) crossing ℓ-regular F Increasing connectivity from ℓ to ℓ+1: the same ratio. (Previous ratio was O(log ℓ) [FL STOC 08]) Open Questions • Can we obtain a constant ratio? Conjecture: No. • Can we please go to dinner?

29. Thanks! Questions?