Paths, Trees and Minimum Latency Tours Kamalika Chaudhuri, Brighten Godfrey, Satish Rao, Satish Rao, Kunal Talwar UC Berkeley

The Problem Given: Given: 1. V: Set of points 2. d : Distance function on pairs of points 3. s : Starting point Find a tour of all points, starting at s, which minimizes the total latency Also called the Traveling Repairman problem Also called the Traveling Repairman problem

Some Results [SG74] NP-Hard on general graphs [SG74] NP-Hard on general graphs [Sitters02] NP Hard on weighted trees [Sitters02] NP Hard on weighted trees [BCCPRS94] MAX-SNP Hard on general graphs [BCCPRS94] MAX-SNP Hard on general graphs [BCCPRS94] Constant factor algorithm for metric spaces [BCCPRS94] Constant factor algorithm for metric spaces [GK96] Approximation ratio : ε [GK96] Approximation ratio : ε Our approximation ratio : 3.59 Our approximation ratio : 3.59

An Algorithm [BCCPRS94] For j=1,2,3,.. For j=1,2,3,.. Find a tree T j of cost at most 2 j which has the most vertices (*) Find a tree T j of cost at most 2 j which has the most vertices (*) Double T j and shortcut to get tour P j Double T j and shortcut to get tour P j Concatenate tours P 1,P 2,… Concatenate tours P 1,P 2,…

Analysis Suppose 2 j · latency OPT (i) · 2 j+1 Suppose 2 j · latency OPT (i) · 2 j+1 T j+1 has at least i vertices T j+1 has at least i vertices Latency of the i th vertex in our tour is at most Latency of the i th vertex in our tour is at most 2 £ 2 j  k 2 k 2 £ 2 j  k 2 k · 8 latency OPT (i) · 8 latency OPT (i) Problem: Problem: Assumed that we can find exact solution to k-MST (the minimum spanning tree with k vertices) Assumed that we can find exact solution to k-MST (the minimum spanning tree with k vertices)  approximate k-MSTs : approximation factor 8   approximate k-MSTs : approximation factor 8 

Trees vs. Paths [BCCPRS94], [GK96] [BCCPRS94], [GK96] Lower bound: k-MST Lower bound: k-MST Tours from k-MSTs Tours from k-MSTs Tours of geometrically increasing lengths Tours of geometrically increasing lengths 3.59  ¼ 7.18 approximation 3.59  ¼ 7.18 approximation Our Algorithm Lower bound: k-stroll Tours from good k-trees Tours of geometrically increasing lengths 3.59 approximation

Trees vs. Paths Our Algorithm Lower bound: k-stroll Tours from good k-trees Tours of geometrically increasing lengths 3.59 Approximation This talk: 1. k-(stroll, tree) 2. Finding good k-trees

Paths vs. Trees Our contribution: Our contribution: Use k-stroll as a lower bound instead of k-MST Use k-stroll as a lower bound instead of k-MST k-stroll : Given s, the minimum cost path from s with k vertices But k-stroll does not seem any easier than k-MST !

Good k-trees Good k-tree : 1. k vertices 2. Tree cost · optimal k-stroll Find a good k-tree by a modification of the k-MST algorithm [Garg96,AK00]

Finding good k-trees [Garg96, AK00] use a variant of the primal-dual algorithm of [GW92] [Garg96, AK00] use a variant of the primal-dual algorithm of [GW92] Allot a budget to each vertex Allot a budget to each vertex Different s produce trees of different size k Different s produce trees of different size k

Finding good trees Our algorithm: Our algorithm: Fix endpoint t Fix endpoint t Budget ∞ to t, λ to all other vertices Budget ∞ to t, λ to all other vertices Run the primal dual algorithm Run the primal dual algorithm This may not give trees for all k This may not give trees for all k Use [Garg96,AK00] to find trees for all k Use [Garg96,AK00] to find trees for all k Argue [ALW02] that we need only the trees produced Argue [ALW02] that we need only the trees produced

Analysis – Basic Ideas Tree LP: min  e c e x e min  e c e x e  e 2  (S) x e ¸ 1 : 8 S ½ V Path LP: min  e c e x e min  e c e x e  e 2  (S) x e ¸ 1 : 8 S ½ V – {s}, t 2 S  e 2  (S) x e ¸ 2 : 8 S ½ V – {s,t}

Analysis – Dual LPs Tree LP: max  S y S max  S y S  S:e 2  (S) y S · c e 8 e Path LP: max 2  S y S -  T:t 2 T y T  S:e 2  (S) y S · c e 8 e

Analysis – Dual LPs Tree LP: max  S y S max  S y S  S:e 2  (S) y S · c e 8 e Path LP: max 2  S y S -  T:t 2 T y T  S:e 2  (S) y S · c e 8 e Tree Primal Cost Tree Dual Cost · 2(1-1/n) £ ¼ Path Dual Cost [GW92] ) Cost of the tree · ·Cost of Opt Path ) Cost of the tree · ·Cost of Opt Path

Analysis – Basic Ideas Tree Primal Cost Tree Dual Cost · 2(1-1/n) £ ¼ Path Dual Cost [GW92] ) Cost of the tree · ·Cost of Opt Path ) Cost of the tree · ·Cost of Opt Path Tree LP: min  e c e x e min  e c e x e  e 2  (S) x e ¸ 1 : 8 S ½ V Path LP: min  e c e x e min  e c e x e  e 2  (S) x e ¸ 1 : 8 S ½ V – {s}, t 2 S  e 2  (S) x e ¸ 2 : 8 S ½ V – {s,t}

Running Time Running Time = O(n 3 log n) Running Time = O(n 3 log n) O(n 2 ) time to run primal dual O(n 2 ) time to run primal dual O(log n) values of O(log n) values of O(n) guesses for t O(n) guesses for t Example shows guessing t appears to be necessary Example shows guessing t appears to be necessary

Conclusion Improved approximation factors for Improved approximation factors for Minimum latency : 3.59 Minimum latency : 3.59 k-Minimum latency: 8.47 k-Minimum latency: 8.47 [GK96] 3.59 is the best we can do by stitching together tours [GK96] 3.59 is the best we can do by stitching together tours Is there an LP based approach which does better? Is there an LP based approach which does better? [FLT02] Better approximation for minimum latency set cover [FLT02] Better approximation for minimum latency set cover