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

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Feb 27, 2024 47 likes •237 views

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 Fabin A. Chudak (Tellabs) David P. Williamson (IBM Almaden) The k-MST

SLIDE 1

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)

SLIDE 2

The k-MST Problem

Given :

An undirect ed gr aph G = (V,E)
cost s ce ≥ 0 on edges
a par amet er k

Goal :

min-cost t ree spanning ≥ k

ver t ices

11 12 5 10 2 1 2 3 2 5 Example (k = 4):

SLIDE 3

Some Hist ory

Fact : k-MST is NP-hard Approximat ion Algorit hms:

O(k ) [Ravi et al. 94]
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]

½

SLIDE 4

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

ce ≥ 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?

SLIDE 5

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 ?

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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 also: T f ails t o span = n-k ver t ices

T

SLIDE 9

Lagrangean Relaxat ion of k-MST

Choose: value f or Lagr angean penalt y par amet er ? ≥ 0 Lagrangean r elaxat ion k-MST(?): 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

# of unspanned vert ices

SLIDE 10

Pr ize-Collect ing St einer Tree

Suppose: penalt y on ever y ver t ex is set t o ? ≥ 0. minimize: c(T) + ? |V\ T| subj ect t o: ver t ices spanned by T ar e connect ed

⇒ only dif f erence f rom k-MST(?) is missing -?(n-k) in obj f n

# of unspanned vert ices same as bef ore

SLIDE 11

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

St ronger [GW 95]:

c(T) + 2 × penalt y of V\ T =2 × dual cost c(T) + penalt y

f V\ T

f rom dual of PCST LP relaxat ion

SLIDE 12

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)

SLIDE 13

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 ?

SLIDE 14

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:

– ?1 < ?2 and ?1 ≈ ?2 – (?1, T1, y1), T1 spans k1 < k ver t ices – (?2, T2, y2), T2 spans k2 > k vert ices

will combine T1, T2 and y1, y2 t o get

f easible + near-opt imal solut ion

SLIDE 15

Combining Two Guarant ees

So f ar: we have guarant ees f or i=1,2:

c(Ti) + 2?i(n-ki) = 2[P CST dual cost of yi]

I dea: t ake a convex combinat ion of t he

t wo so previous calculat ion works. ⇒ choose µ1, µ2 s.t . µ1 (n-k1) + µ2 (n-k2) = n-k ⇒ assume ?=?1 =?2, get : µ1 c(T1) + µ2 c(T2) + 2?(n-k) = 2[P CST dual cost of µ1 y1 + µ2 y2]

Result : µ1 c(T1) + µ2 c(T2) = 2 × OP

T

f easible dual

SLIDE 16

An Easy Case

We have: µ1 c(T1) + µ2 c(T2) = 2OP T T2 (t he big t r ee) is a f easible solut ion

⇒ if µ2 ≥ ½

, j ust r et ur n T2 f or a 4-appr oximat ion ⇒ can assume µ1 ≥ ½ , µ2 =½ I n har der case: supplement t he small t ree (T1) wit h a f ew ver t ices f r om T2

SLIDE 17

Supplement al Vert ices

Algor it hm [Gar g 96]

T2 T1 double edges of T2 + short cut t o t our: connect T1 t o cheapest pat h of t our: k = 7 k1 = 4, k2 = 10

SLIDE 18

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

f 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