Augmenting Paths, Witnesses and Improved Approximations for Bounded Degree MSTs

Augmenting Paths, Witnesses and Improved Approximations for Bounded Degree MSTs K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar UC Berkeley

BDMST : The Problem • Given: • Graph: G • Edge costs: ce • Degree bound : D • Find a minimum cost tree that respects the degree bounds 2 2 1 1 1 2 2 1 1 1 2 2 MST BDMST, D = 3

BDMST : The Problem • Generalization of Minimum Cost Hamiltonian Path • For general weighted graphs, • No Polynomial-Factor Approximation unless P=NP • Our Work: • Relax degree bounds to obtain an approximation in cost

Previous Results • [FR94] Unweighted graphs • Additive 1 approximation to degree • [KR00] Weighted graphs, uniform degree bounds • deg(v) · b(1+) D + logb n • cost(T) · (1 + 1/) OPT • [KR03] Non-uniform degree bounds • [CRRT] Quasipolynomial Running Time • deg(v) · D + log n / log log n • cost(T) · OPT • [CRRT]Polynomial Running Time • deg(v) · bD + logb n • cost(T) · OPT

LP Formulation • Primal: min e ce xe e 2(v) xe· D x 2 SPG • Dual: maxv minT (C(T) + vv(degT(v) - D)) MST in cost function cuv + u + v v : Penalties on high-degree vertices

An Algorithm • Solve Dual LP • Optimal Penalties v • Pick MST in cost function (cuv + u + v) with: • Low maximum degree • Low actual cost

An Algorithm • Solve Dual LP • Optimal Penalties v • Pick MST in cost function (cuv + u + v) with: • Maximum degree : D • Real cost : OPT

An Algorithm • Solve Dual LP • Optimal Penalties v • Pick MST in cost function (cuv + u + v) with: • Maximum degree : D + O(log n/loglog n) • Real cost : OPT

Picking the Right Tree • T is MST in cost function cuv + u + v with: • Max degree : D + O(log n/loglogn) • For every vertex v with v > 0, Min degree : D – O(log n/loglog n) • Theorem: T has • Max degree: D + O(log n/log log n) • Actual cost : OPT

MSTDB Problem • Given: • Graph: G • Edge costs: ce • Degree upper bound: DH • A set of nodes: L • Degree lower bound on L: DL • Find a MST with • Max degree : DH • Min degree of L : DL • Or prove that no such tree exists 2 1 1 1 1 2 1 1 1 1 1 2 DH = 3, DL = 2, L = O

Previous Work • [FR94] Unweighted graphs, degree upper bounds • Additive 1 approximation • [F93] Degree upper bounds • Finds an MST with max degree bD + logb n • Or proves no MST with max degree D exists

Our Guarantees • Given: • Graph: G • Edge costs: ce • Degree upper bound: DH • A set of nodes: L • Degree lower bound on L: DL • We can find an MST with: • Max Degree: DH + O(log n/log log n) • Min Degree of L : DL – O(log n/log log n) • Or prove that no MST with given bounds exist

Final BDMST Algorithm • Solve Dual LP • Optimal Penalties v • Run our MSTDB algorithm : • Cost function : cuv + u + v • Degree upper bound : D • L: Set of nodes with v > 0 • Degree lower bound : D • Theorem: Failure contradicts optimality of v

MSTDB Algorithm Outline • Start with arbitrary MST T0 • Phase i: Ti-1 : use Augmenting Paths to • Reduce the degree of a “high degree” node • Or increase the degree of a “low degree” node • Success: • New tree Ti • Failure: • Witness for either DH = dmax(Ti-1) – O(log n/log log n) or DL = dmin(L) + O(log n/log log n)

1 B A 1 1 1 D 2 C Useful Edges and Witness Witness: Structure to show a lower(upper) bound on the degree of any MST • Useful edge: • Occurs in some MST of G e f

High Degree Witness • High degree Witness: • Center Set : W • Clusters: C1, .., Ck • No useful intercluster edge W In any MST: Max Degree (W) ¸d (|W| + k – 1) / |W| e

Feasible Swaps • Feasible swap (e,f,T): • Tree edge e • Non-tree edge f • Unique cycle in T [ f contains e • c(e) = c(f) • A feasible swap (e,f,T): • Produces an equal cost tree • Reduces degree of endpoints of e 1 B A 1 1 1 e f D 2 C

Augmenting Paths • Algorithm: • Construct an augmenting path of feasible swaps

Degree d – 1 node Degree d node Tree edge Low degree node Useful nontree edge W 2 2 1 1 Ci 1 1

Algorithm Outline • Start with: • Center Set : W0 • Clusters : connected through W0 • In step i: • Find an augmenting path of feasible swaps to improve some v in Wi • Failure: Center Set and clusters form a high degree witness

Conclusion • Improved approximation for: • BDMST (Bounded Degree MST) • MSTDB (MST with Degree Bounds) • New Techniques: • Improved cost bounding techniques based on Edmond’s matching algorithm • Improved degree improvement techniques based on augmenting paths • Open Question: • Can degree bounds be relaxed to additive constant? • [FR94] Gives additive 1 for unweighted graphs

Questions?

Reduce-Max-Degree Initialize: • Let • Sd = nodes with degree d or more • Pick d : • |Sd-1| · (log n/log log n) |Sd| • Center Set : • W0 = Sd[ Sd-1 • Initial clusters: • Components when W0 is deleted from T

Degree d – 1 node Degree d node Tree edge Low degree node Useful nontree edge 1 1 1 1 W 2 2 Ci

Reduce-Max-Degree • For each useful intercluster edge f : • Find feasible swap (e,f,T) that improves v 2 Wt • Remove v from Wt • Form new cluster with: • e • f • v • children clusters of v

Degree d – 1 node Degree d node Tree edge Low degree node Useful nontree edge 1 1 1 1 2 2

Reduce-Max-Degree Termination Conditions: • Degree d vertex v removed from Wt: • Can find a sequence of swaps to improve v • Number of degree d vertices decreases by one

Reduce-Max-Degree Termination Conditions: • Degree d vertex v removed from Wt: • Can find a set of swaps which improve v • Number of degree d vertices decreases by one • No feasible intercluster edges: • Can find witness to show that max degree of any MST is at least d – O(log n/log log n)

Obtaining a Witness • Center Set : Wt • |Wt| · |Sd| • Clusters: • From deleting Wt: ¸ (d-2)|Wt| • Lost from merges: · |Sd-1| • Total: ¸ (d-2)|Wt| - |Sd-1| • Witness Quality: • At least (d-2) - O(log n/log log n)

Summary • Improved approximation for: • BDMST (Bounded Degree MST) • MSTDB (MST with Degree Bounds) • New Techniques: • Improved cost bounding techniques based on Edmond’s matching algorithm • Improved degree improvement techniques based on augmenting paths • Open Question: • Can degree bounds be relaxed to additive constant? • [FR94] Gives additive 1 for unweighted graphs

A Better Algorithm • Suppose given DH, we can find an MST with : • Max Degree : 5DH + 2 • Or show there is no MST with max degree DH • BDMST Guarantees: • deg(v) · 10(1+)D + O(1) • cost(T) · (1+1/) OPT • MSTDB Algorithm uses Push-Relabel

Push Relabel[G85, GT86] • A node has: • A Label • An Excess • Push: • Push flow from a higher to a lower label along an edge • Relabel: • Raise the label of a node with no edges to a lower label • Feasibility: • Node at label L has edges to nodes at label L-1 or above

Push Relabel for Max Flow • Initially: • Source : 1 excess • Sink : 1 deficit • All other nodes: 0 excess • Push and Relabel until: • No node has any excess • Or there is a label with no nodes A

Push-Relabel for MSTDB • Works for MSTDB with degree upper bounds only • Each node has: • A label • An excess/deficit degree • Excess and Deficits: • deg(v) ¸ d : (deg(v) – d + 1) units excess • deg(v) < d-1: (d – 1 – deg(v)) units deficit

Push-Relabel for MSTDB • Push: • A node can transfer degree to a node at a lower label • Relabel: • Raise the label of a node which cannot transfer degree to any node at a lower level • Feasibility: • A node at label L can be improved only by nodes at label L-1 or higher

Algorithm Outline • Sparse Label: • Has less than 4 times as many nodes as the number of nodes in all the labels above it • Algorithm: Push and relabel until: • Either no nodes have any excess • Or there is a sparse label • Sparse Label ! Witness • Average degree : d/5 - 2

Publications & Manuscripts: • [CRRT] Improved Approximation Algorithms for Degree Bounded MSTs using Push-Relabel • [CRRT] What would Edmonds do? Augmenting Paths and Witnesses for Degree Bounded MSTs • [CCWBPK04] Selfish Caching in Distributed Systems : A Game Theoretic Approach, PODC 2004 • [CGRT03] Paths, Trees and Minimum Latency Tours, FOCS 2003

Augmenting Paths, Witnesses and Improved Approximations for Bounded Degree MSTs