Approximating Node-Weighted Survivable Networks Zeev Nutov The Open University of Israel

2 Talk Outline Problem Definition History and Our Results Greedy Algorithm for Node Weighted Steiner Trees Reducing NWSN to Finding Minimum Weight Edge-Cover of Uncrossable Set-Family Spider-Cover Decomposition of Edge-Covers of Uncrossable Set-Families Algorithm for Covering Uncrossable Set-Families Node-Weighted k-Flow is harder than Densest -Subgraph

3 Survivable Network (SN) Instance: A graph G = (V,E), weight function w on edges/nodes, U  V, and connectivity requirements r(u,v), u,v  U. Objective: A minimum weight spanning subgraph J of G containing U so that λ J (u,v) ≥ r(u,v) for all u,v  U λ J (u,v) = uv-edge-connectivity in J Problem Definition Approximability Edge-weights: 2-approximable [Jain, FOCS 98], APX-hard Node-weights: for r(u,v)  {0,1} O(log n)-approximable, Set-Cover hard [Klein and Ravi, IPCO 93]

4 History and Our Results Edge-weightsYearNode-weights 2 for r(u,v)  {0,1} [AKR] r max [WGMV] H(n) for r(u,v)  {0,1} [KR] 2H(r max ) [GGPSTW] H(n) for r(u,v)  {0,1}[GK] 2 [J] 1998 Theorem 1 NWSN admits a r max · 3H(n) - approximation algorithm. Theorem 2 ρ-approximation for NWSN with |U|=2 implies 1/ρ 2 - approximation for Densest -Subgraph. We do not have a polylogarithmic approximation for any r max … But this is not our fault! What about node-weights and r max =2?

5 Node Weighted Steiner Tree Instance: A graph G=(V,E), a set U  V of terminals, weights w(v) for nodes in V−U. Objective: Find a min-weight subtree T of G containing U. 5 a 2 c b d The “deficiency” of a partial solution I: v(I) = # (components containing terminals in (V,I)) -1. v(I) = 0 w(I) = 5 v(I) = 0 w(I) = 8 v(I) = 1 w(I) = 7 The “node-weight” w(I) of a partial solution I  E: w(I) = w(V(I)) = the weight of endnodes of I

6 The Greedy Algorithm Initialize: I   While ν(I) > 0 do: Find S  E – I so that I  I  F Return I. The Density Condition Theorem: If ν is decreasing and w is subadditive then the greedy algorithm has approximation ratio ρ ·H(ν(  )). Objective: Find in polynomial time an “augmentation” S that satisfies the density condition for “small” ρ.

7 A Lesson in Zoology These are also spiders: In general, a spider is a tree on at least 2 nodes, which has at most one node of degree ≥ 3. This is a spider:

8 Spider Decomposition of Trees Center – The single node of degree ≥ 3. If there is no node of degree ≥ 3, any node can be a center. Leaves – The non-center nodes of degree 1. Lemma: Every tree can be decomposed into node-disjoint spiders such that every leaf of the tree belongs to a unique spider. 1.Select a node v whose sub- tree is a spider. 2.Remove v and its sub-tree. 3.Remove the path from v to its closet ancestor of degree  3. 4.Repeat.

9 Finding the First Augmentation Finding a spider S (in fact, a Shortest Path Tree) of optimal density: For each node s in the graph 1.Sort the paths from s to terminals in increasing weight order. 2.Add the two lightest paths. 3.Add paths in increasing weight order, till reaching minimum density Terminals: 2 Weight: 8 Density: 8 = 8/(2-1) Terminals: 4 Weight: 19 Density: ~ 6.3=19/(4-1) T = optimal tree; we may assume: terminals = leaves of T  By averaging, there is a spider S i such that: Terminals: 3 Weight: 12 Density: 6 = 12/(3-1) {S i } – spider-decomposition of T. The spiders are disjoint  3

10 The Complete Algorithm Finding an augmentation with a general partial cover I: 1.Contract every connected component of (V,I) into a super-node; a super node is a super-terminal if it contains a terminal. 2.Find an augmentation in the new graph (the partial cover is now  ). The previous algorithm finds an augmentation obeying the Density Condition with ρ=2 if the current partial cover is I = . The approximation ratio of the algorithm is 2H(|U|).

11 Algorithm for NWSN The algorithm has r max iterations. In iteration k we find a 3H(n)-approximation for the problem: Given: A graph J=J k-1 with λ J (u,v) ≥ min{r(u,v),k-1} for all u,v  U Find: An edge set I with w(V(I)) minimum so that λ J+I (u,v) ≥ min{r(u,v),k} for all u,v  U Hence after r max iterations, a feasible solution of weight at most r max ·3H(n)·opt is found. Instance: A graph G = (V,E), weight function w on the nodes, U  V, and connectivity requirements r(u,v), u,v  U. Objective: A minimum weight spanning subgraph J of G containing U so that λ J (u,v) ≥ r(u,v) for all u,v  U.

12 Covers of Uncrossable Set-Families Node-Weighted Set-Family Edge-Cover (NWSFC) The augmentation problem we want to solve is a particular case of the following problem: Instance: A graph (V,E), node weights {w(v):v  V}, and an uncrossable set-family F on V. Objective: Find an F -cover I ⊆ E of minimum node-weight (edge e covers set X if e has exactly one endnode in X) F is uncrossable if X,Y  F implies at least one of the following: or X ∩Y, X  Y  F X − Y, Y −X  F X YV−Y V−X Note: The inclusion minimal members of F are pairwise disjoint.

13 Spider-Covers of Uncrossable Set-Families C ( F ) = the family of inclusion minimal sets in F (min- cores) F (C) = sets in F that contain a unique min-core C (cores) F (s,C) = {X  F (C) : s  V-X} F (s, C ) =  { F (s,C) : C  C } Definition: Let C ⊆ C ( F ) and let s  V. An edge set S is an F (s, C )-cover if: -S covers F (s,C) for every C  C -if C ={C} then no member of F (C) contains s An F (s, C )-cover S is a spider-cover if it can be partitioned into F (s,C)-covers {S C :C  C } so that the node sets {V(S C ) −s} are pairwise disjoint.

14 Spider-Coves Decompositions Definition of a Spider-Cover Decomposition: A sub-partition S 1,…,S q of a cover I is a spider cover decomposition of I if there exists a partition C 1, …, C q of C ( F ) and centers s 1,…,s q  V so that: -Each S i is an F (s i, C i )-cover - The node sets V(S i ) are pairwise disjoint. Spider-Cover Decomposition Theorem: Any uncrossable family cover has a spider-cover decomposition. Proof: Later.

15 Covering Uncrossable Families S is a spider with leaves: Δ(S) ≥ /2 (tight for =2) S is an F (s, C )-cover with | C |= : Δ(S) ≥ /3 (tight for =3) = | C ( F )| = # (min-cores) Δ(S) = decrease in the deficiency caused by adding S to the partial solution Density Condition (for I=  ) If S is an F (s, C )-cover then Δ(S) ≥ (| C |-1)/2 if | C | ≥2 Δ(S) = 1 if | C | =1 The Spider-Cover Lemma Tight Example

16 The Algorithm Thus the Greedy Algorithm can be implemented in polynomial time with ρ=3. Approximation ratio: 3H( (  )) = 3H(| C ( F )|) ≤ 3H(n) The Spider-Cover Lemma implies that there exists a spider- cover that satisfies the Density Condition with ρ=3. Such spider-cover can be found in polynomial time assuming we can compute in polynomial time: -The family C ( F ) of min-cores (max-flows) -Minimum weight F (s,C)-cover (min-cost k-flows)

17 The Spider-Cover Decomposition Thm – Proof Sketch We may assume that I is an inclusion minimal F -cover. Then for every e  I there exists a witness set W e  F, namely: e is the unique edge in I that covers W e. A family W = {W e : e  I} is called a witness family for I (every e  I has a unique witness set in W e  W ). Notation F – uncrossable family ( X,Y  F implies X ∩Y, X  Y  F or X−Y,Y−X  F ) I – an F -cover (for any X  F there is e  I with exactly one endnode in X) Lemma: Let I be an inclusion minimal cover of an uncrossable family F. Then there exists a witness family L for I which is laminar.

18 The Spider-Cover Decomposition Thm – Proof Sketch For a min-core C  C ( F ) define: L C = the maximal set in L containing C e C = the unique edge in I covering L C, e C =s C v C, v C  L C S C = edges in I contained in L C plus e C Assumptions: – Every member of F is a core – Every minimal member (leaf) of L is a min-core.

19 The Spider-Cover Decomposition Thm – Proof Sketch Lemma: The sets {L C : C in C ( F )} are pairwise disjoint. The sets {S C : C in C ( F )} are pairwise disjoint. S C covers all cores contained in L C. Corollary: Any partition C 1, …, C q of C ( F ) induces a partition S 1,…,S q of I. We seek a partition so that S 1,…,S q is a spider-cover decomposition. - A natural partition of C ( F ) is by the stars of {e C : C  C ( F )}. - This approach fails for 1-edge stars; S C is not a spider-cover if there is a dangerous set M C  F containing L C +s C - Every star with at least 2 edges indeed induces a spider-cover.

20 The Spider-Cover Decomposition Thm – Proof Sketch How do we group dangerous cores? - group some together, or - assign to “non-dangerous” stars. Assigning singleton classes: Every singleton class {M C } of R is assigned to the part of any edge e C’ covering M C. Observation: Every dangerous M C is covered by some edge e C’. Grouping dangerous cores together: The relation R ={(C,C’) : M C ∩ M C’ ≠  } is an equivalence, and its classes of size ≥2 induce spider-covers (center − any node in the intersection of M C ’s)

21 B A s t I Reducing NWSN to bipartite D S

22 Summary and Open Questions What did we do? Generalized the decomposition of a tree into spiders to covers of uncrossable families (looks easy after found…) What do we get? E.g., an r max ·3H(n)-approximation algorithm for NWSN. Any other applications? Probably YES. Open Question: Node-Weighted k-Flow (NWkF) is a special case of NWSN where r(s,t)=k and r(u,v)=0 otherwise. NWkF admits a k-approximation algorithm. Anything better, even for unit weights?