Approximat e k-MSTs and k-St einer Trees via t he Primal-Dual Met - PDF document

Approximat e k-MSTs and k-St einer Trees via t he Primal-Dual Met hod and Lagrangean Relaxat ion Tim Roughgarden Cornell Universit y and I BM Almaden joint work with Fabián A. Chudak (Tellabs) David P. Williamson (IBM Almaden)

The k-MST Problem Given : • An undirect ed gr aph G = (V,E) • cost s c e ≥ 0 on edges • a par amet er k Goal : • min-cost t ree spanning ≥ k ver t ices 11 Example (k = 4): 5 2 3 2 1 2 10 5 12 2

Some Hist ory Fact : k-MST is NP-hard Approximat ion Algorit hms: ½ • O(k ) [Ravi et al. 94] 2 • O(log k) [Awerbuch et al. 95] • O(log k) [Raj agopalan/ Vazirani 95] • 17 [Blum/ Ravi/ Vempala 95] • 5 [Gar g 96] • 3 [Gar g 96] • 2 + ? [Arora/ Karakost os 00] • 2 [Gar g 00] 3

Mot ivat ing Quest ion Observat ion: all const ant - f act or appr ox algs f or k-MST rely on a primal-dual alg f or prize-collect ing St einer t ree Pr ize-collect ing St einer t ree: • given: graph G=(V,E), cost s c e ≥ 0 on E, penalt ies ≥ 0 on V • goal: t r ee minimizing cost of it s edges + penalt ies of unspanned vert ices Quest ion: why? 4

The Connect ion Punchline : a PCST pr oblem ar ises as a Lagr angean r elaxat ion of t he k-MST problem Roughly: • complicat ing const raint = t r ee spans ≥ k ver t ices • lif t t o obj ect ive f unct ion – use paramet er ? ≥ 0 t o penalize t rees spanning < k ver t ices ⇒ get a P CST problem wit h all vert ex penalt ies equal t o ? 5

The Agenda Our goal: wit h t his insight , revisit exist ing appr ox algs f or k-MST and derive • a simpler algor it hm descr ipt ion • a simpler pr oof of appr ox r at io ⇒ we will f ocus on Garg’s 5- approximat ion algorit hm 6

Our I nspirat ion J ain/ Vazir ani ‘99 gave: • a primal-dual 3-appr oximat ion algorit hm f or uncapacit at ed f acilit y locat ion • a r educt ion f r om k-median t o f acilit y locat ion, via Lagr angean r elaxat ion – nast y const raint = open = k medians – f acilit y cost = penalt y paramet er ? ⇒ yields a 6-approximat ion algorit hm f or k-median 7

Sket ch of k-MST Formulat ion minimize: cost of edges in t ree T = c(T) subj ect t o: ver t ices spanned by T ar e connect ed T also: T f ails t o span = n-k ver t ices 8

Lagrangean Relaxat ion of k-MST Choose: value f or Lagr angean penalt y par amet er ? ≥ 0 Lagrangean r elaxat ion k-MST(?): # of unspanned vert ices minimize: c(T) + ? [|V\ T| - (n-k)] subj ect t o: ver t ices spanned by T are connect ed Fact : ? ≥ 0 ⇒ lower bound f or k-MST 9

Pr ize-Collect ing St einer Tree Suppose: penalt y on ever y ver t ex is set t o ? ≥ 0. # of unspanned vert ices minimize: c(T) + ? |V\ T| subj ect t o: ver t ices spanned by T ar e connect ed same as bef ore ⇒ only dif f erence f rom k-MST(?) is missing -?(n-k) in obj f n 10

A Primal-Dual Algorit hm f or P CST Good news: we know how t o appr oximat e PCST well. • Theorem [Goemans, Williamson 95]: I n poly-t ime, can const ruct a t ree T + f easible (f ract ional) dual s.t . primal cost = 2 × dual cost c(T) + penalt y f rom dual of PCST of V\ T LP relaxat ion • St ronger [GW 95]: c(T) + 2 × penalt y of V\ T =2 × dual cost 11

From PCST t o k-MST $64K Quest ion: What does t he PCST result imply f or k-MST? • int erpret k-MST(?) as a PCST inst ance wit h all penalt ies = ? ≥ 0 Problems: • dif f er ent pr imal obj ect ive f ns – k-MST: c(T) – PCST: c(T) + ? | V\ T| • dif f er ent duals – recall: k-MST(?), PCST dif f er only by a ?(n-k) t erm in obj f ns ⇒ duals ident ical except f or t his ?(n-k) t erm (in obj f ns) 12

An I nherit ed Guarant ee • running GW algorit hm on k-MST(?) yields t ree T, dual s.t .: c(T) + 2? | V\ T| = 2 × P CST dual cost • if | V\ T| = n-k (T spans k vert ices) ⇒ subt ract ing 2?(n-k) on each side: c(T) = 2 × [P CST dual cost - ?(n-k)] = 2 × [k-MST(?) dual cost ] = 2 × OP T ⇒ done if we can f ind magic value f or ? 13

The Cat ch Problem: what if no value of ? yields a t ree T spanning exact ly k ver t ices? Solut ion (a la [J ain/ Vazirani 99]) • ? = 0 ⇒ T will be empt y • ? suf f . large ⇒ T spans all vert ices • via bisect ion search, can f ind: ? 2 and ? 1 ≈ ? 2 – ? 1 < – (? 1 , T 1 , y 1 ), T 1 spans k 1 < k ver t ices – (? 2 , T 2 , y 2 ), T 2 spans k 2 > k vert ices • will combine T 1 , T 2 and y 1 , y 2 t o get f easible + near-opt imal solut ion 14

Combining Two Guarant ees So f ar: we have guarant ees f or i=1,2: c(T i ) + 2? i (n-k i ) = 2[P CST dual cost of y i ] I dea: t ake a convex combinat ion of t he t wo so previous calculat ion works. ⇒ choose µ 1 , µ 2 s.t . µ 1 (n-k 1 ) + µ 2 (n-k 2 ) = n-k ⇒ assume ?=? 1 =? 2 , get : f easible dual µ 1 c(T 1 ) + µ 2 c(T 2 ) + 2?(n-k) = 2[P CST dual cost of µ 1 y 1 + µ 2 y 2 ] Result : µ 1 c(T 1 ) + µ 2 c(T 2 ) = 2 × OP T 15

An Easy Case We have: µ 1 c(T 1 ) + µ 2 c(T 2 ) = 2OP T T 2 (t he big t r ee) is a f easible solut ion ⇒ if µ 2 ≥ ½ , j ust r et ur n T 2 f or a 4-appr oximat ion ⇒ can assume µ 1 ≥ ½ , µ 2 =½ I n har der case: supplement t he small t ree (T 1 ) wit h a f ew ver t ices f r om T 2 16

Supplement al Vert ices k = 7 Algor it hm [Gar g 96] k 1 = 4, k 2 = 10 T 1 T 2 double edges of T 2 + short cut t o t our: connect T 1 t o cheapest pat h of t our: 17

k-St einer Trees The k-St einer t r ee pr oblem: • given a gr aph wit h cost s on edges and a dist inguished set of t er minal vert ices • f ind t he min-cost t ree spanning at least k t er minals Result : • can ext end pr evious algor it hm + analysis, get a 5-appr oximat ion – use vert ex penalt ies only f or t erminals 18

Open Quest ions Direct ions f or f ut ure work: • f ur t her applicat ions of J ain/ Vazir ani’s t echniques – Garg’s 3-approximat ion algorit hm – ot her NP-hard problems • ext end f r amewor k t o handle many complicat ing const r aint s 19