Approximating Node-Weighted Survivable Networks

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

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 l-Subgraph

Problem Definition 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 Jof G containing U so that λJ(u,v) ≥ r(u,v) for all u,vU λJ(u,v) =uv-edge-connectivity in J 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]

Theorem 1 NWSN admits a rmax·3H(n)-approximation algorithm. Theorem 2 ρ-approximation for NWSN with |U|=2 implies 1/ρ2-approximation for Densest l-Subgraph. History and Our Results What about node-weights and rmax =2? We do not have a polylogarithmic approximation for any rmax… But this is not our fault!

4 4 d a 1 5 5 3 3 3 3 c 2 2 b 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. v(I) = 0 w(I) = 8 v(I) = 1 w(I) = 7 v(I) = 0 w(I) = 5 The “node-weight” w(I)of a partial solutionI  E: w(I) = w(V(I)) = the weight of endnodes of I The “deficiency”of a partial solution I: v(I) = # (components containing terminalsin (V,I)) -1.

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.

This is a spider: 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.

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. • Select a node v whose sub-tree is a spider. • Remove v and its sub-tree. • Remove the path from v to its closet ancestor of degree  3. • Repeat.

2 4 7 3 Finding the First Augmentation T = optimal tree;we may assume: terminals = leaves of T {Si} – spider-decomposition of T. The spiders are disjoint   By averaging, there is a spider Si such that: Finding a spider S (in fact, a Shortest Path Tree) of optimal density: • For each node s in the graph • Sort the paths from s to terminals in increasing weight order. • Add the two lightest paths. • Add paths in increasing weight order, till reaching minimum density. 3 Terminals: 2 Weight: 8 Density: 8 = 8/(2-1) Terminals: 3 Weight: 12 Density: 6 = 12/(3-1) Terminals: 4 Weight: 19 Density: ~ 6.3=19/(4-1)

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

Algorithm for NWSN 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 Jof G containing U so that λJ(u,v) ≥ r(u,v) for all u,vU. The algorithm has rmax iterations. In iteration k we find a 3H(n)-approximation for the problem: Given: A graph J=Jk-1 with λJ(u,v) ≥ min{r(u,v),k-1} for all u,vU Find: An edge set I with w(V(I)) minimumso that λJ+I(u,v) ≥ min{r(u,v),k} for all u,vU Hence after rmax iterations, a feasible solution of weight at most rmax ·3H(n)·opt is found.

Covers of Uncrossable Set-Families The augmentation problem we want to solve is a particular case of the following problem: Node-Weighted Set-Family Edge-Cover (NWSFC) 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 ⊆ Eof 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: Y V−Y X ∩Y, XUYF or X X − Y, Y −X F V−X Note: The inclusion minimal members of F are pairwise disjoint.

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) = U{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 {SC:CC} so that • the node sets {V(SC) −s} are pairwise disjoint.

Spider-Coves Decompositions • Definition of a Spider-Cover Decomposition: • A sub-partition S1,…,Sq of a cover I is a spider cover • decomposition of I if there exists a partition C1, …,Cq of • C(F) and centers s1,…,sqV so that: • Each Si is an F(si,Ci)-cover • - The node sets V(Si) are pairwise disjoint. Spider-Cover Decomposition Theorem: Any uncrossable family cover has a spider-cover decomposition. Proof: Later.

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

The Algorithm 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) Thus the Greedy Algorithm can be implemented in polynomial time with ρ=3. Approximation ratio: 3H(()) = 3H(|C(F)|) ≤3H(n)

The Spider-Cover Decomposition Thm – Proof Sketch Notation F – uncrossable family (X,YF implies X ∩Y, XUY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) We may assume that I is an inclusion minimal F-cover. Then for every eI there exists a witness setWe F, namely: e is the unique edge in I that covers We. A family W = {We: e  I} is called a witness family for I (every eI has a unique witness set in We  W). 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.

The Spider-Cover Decomposition Thm – Proof Sketch Assumptions: – Every member of F is a core – Every minimal member (leaf) of L is a min-core. • For a min-core CC(F) define: • LC= the maximal set in L containing C • eC= the unique edge in I covering LC, eC=sCvC, vC  LC • SC = edges in I contained in LC plus eC

The Spider-Cover Decomposition Thm – Proof Sketch • Lemma: • The sets {LC : C in C(F)} are pairwise disjoint. • The sets {SC : C in C(F)} are pairwise disjoint. • SC covers all cores contained in LC. Corollary: Any partition C1, …, Cq of C(F) induces a partition S1,…,Sq of I. We seek a partition so that S1,…,Sqis a spider-cover decomposition. - A natural partition of C(F) is by the stars of {eC : CC(F)}. - Every star with at least 2 edges indeed induces a spider-cover. - This approach fails for 1-edge stars; SC is not a spider-cover if there is a dangerous set MC F containing LC+sC

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

Reducing NWSN to bipartite DlS s A I B t

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 rmax·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?

Questions?

Approximating Node-Weighted Survivable Networks