Covering Crossing Biset-Families by Digraphs Zeev Nutov The Open University of Israel

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

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

Intersecting and crossing set-families

Two sets X,Y on a groundset V: 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-family if X ∩ Y, X ⋃ Y  F for any intersecting/crossing X,Y  F. A set-family F is an intersecting/crossing set-family if X ∩ Y, X ⋃ Y  F for any intersecting/crossing X,Y  F.

A directed edge covers a set S if it goes from S to V - S. The Set-Family Edge-Cover problem S V-SV-SV-SV-S 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. 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.

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. Both problems are particular cases of Set-Family Edge-Cover: Rooted Edge-Connectivity Augmentation – intersecting F. Global Edge-Connectivity Augmentation – crossing F. 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.

Examples-cont. Intersecting families - arise in rooted connectivity problems. 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 : d G (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 : d G (S) =ℓ } is crossing. d G (S) = the number of edges in G leaving S.

Approximability of Set-Family Edge-Cover Set-Family Edge-Cover with intersecting F can be solved in polynomial time via a primal-dual algorithm [Frank 1999]. 1.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 % The family F + and the family {V-S : S  F − } of F − -complements are both intersecting. F − -complements are both intersecting. 2.Return J = J + ⋃ J − ; J + is an optimal edge-cover of F + ; J + is an optimal edge-cover of F + ; J − is an optimal “reverse edge-cover” of F − -complements. J − is an optimal “reverse edge-cover” of F − -complements. What about Set-Family Edge-Cover with crossing F ? Can be decomposed into two problems of covering an intersecting set-family; thus admits a 2-approximation.

Intersecting and crossing biset-families

Bisets A biset is an ordered pair of sets S=(S I,S O ) with S I  S O ; S I is the inner part and S O is the outer part of S. Intersection and the union of bisets X,Y are defined by: X ∩ Y = (X I ∩ Y I, X O ∩ Y O ) X ⋃ Y = (X I ⋃ Y I, X O ⋃ Y O )

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

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

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.

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 = {(S I,S O ): S I  S O  V - r, γ (S) + d G (S) = ℓ } F = {(S I,S O ): S I  S O  V - r, γ (S) + d G (S) = ℓ } Crossing families - arise in global connectivity problems. Let G be a directed graph that is ℓ -connected (has ℓ node-disjoint dipaths between any two nodes). Then the following bifamily (“violated” bisets) is crossing F = {(S I,S O ): S I  S O  V, γ (S) = ℓ, d G (S) = 0} F = {(S I,S O ): S I  S O  V, γ (S) = ℓ, d G (S) = 0} SISISISI SOSOSOSO r

Approximability of Bifamily Edge-Cover Bifamily Edge-Cover with intersecting F can be solved in polynomial time via a primal-dual algorithm [Frank 2009]. Can we get a better ratio? What about Bifamily Edge-Cover with crossing F ? 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 }.

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 {S I :S ∈ F }) τ = the optimal value of a natural LP-relaxation

Almost constant approximation 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 A bifamily F is ℓ -regular if γ (S) = ℓ for all S ∈ F and γ (X ∩ Y) ≥ ℓ for any intersecting X,Y ∈ F.

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 F J be the “residual bifamily” of F w.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 ( F J ) ≤ ν ( F )/2.

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

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

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

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

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

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

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

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

Thanks! Questions?