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Universal Approximations for TSP, Steiner Tree, and Set CoverPowerPoint Presentation

Universal Approximations for TSP, Steiner Tree, and Set Cover

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### Universal Approximations forTSP, Steiner Tree, and Set Cover

Authors: Lujun Jia, Guolong Lin, Guevara Noubir,

Rajmohan Rajaraman, Ravi Sundaram

Presented by Songjian Lu

contents

- Introduction
- Definition of UTSP,UST,USC
- Detail of UTSP
- Other result

Introduction

- Consider a courier who delivers packages to different houses and businesses in a city every day. One challenge faced by the courier is to determine a suitable route every day, given the packages to be delivered that day.

UTSP

- Universal traveling salesman problem
- DEFINITION:(UTSP).An instance of UTSP is a metric space (V,d). For any cycle(tour) C containing all the vertices in V and a subset S of V, Let CS denote the unique cycle over S in which the ordering of vertices in S is consistent with their ordering in V. The stretch of C is defined as maxSV ||Cs||/||OptTrS||, where OptTrS denotes the minimum cost tour on set S. The universal traveling salesman problem is to find a tour on V with minimum stretch.

UST

- Universal steiner trees
- Definition:(UST) An instance of the Universal steiner tree problem is a triple <V,d,r>, where (V,d) forms a metric space, and r is a distinguished vertex in V that we refer to as the root. For any spanning tree T of V, define the stretch of T as maxSV||Ts+r||/||OptSts+r, The goal of the UST problem is to determine a spanning tree with minimum stretch.

USC

- Universal Set Cover
- Definition:(USC). An instance of USC is a triple <U,C,c>, where U={e1,e2,…,en} is a ground set of elements, C={S1,S2,…,Sm} is a collection of sets, and c is a cost function mapping C to Q+. We define an assignment f as a function from U to C that satisfies ef(e) for all e in U. We extend the definition of f to apply to any SU as follows: f(S)={f(e):e S}. We next define the cost of f(S), ||f(S)||, as SU c(x) for all x in f(S), and the stretch of an assignment f as max SU c(f(S))/Opt(S), where Opt(S) is the cost of the optimal set cover solution to S. And the goal is to compute an assignment f with minimum stretch.

UTSP detail-1

- Definition: ((r,,I)-PARTITION). A (r,,I)-partition of a metric space (V,d) is a partition {Si} of V such that (i) the diameter of every set Si in the partition is at most r and (ii) for every node vV, the ball Br(v) intersects at most I set in the partition, where Br(v)={uV|d(u,v)r}.
- A (,I)-partition scheme is a procedure that computes a (r,,I)-partition for any r>0.

UTSP detail-2

- Lemma 1: The general metric spaces has a (r,log(n),log(n))-partition that can be computed efficiently for any r>0.
- Lemma 2: For a doubling metric space (V,d) with doubling constant , a (r,1,3)-partition can be computed efficiently for any r>0.

UTSP detail-3

- Lemma 3: If the points are in k-dimensional Euclidean space, a (r,k1/2(2+),2k)-partition can be computed efficiently for any r>0, >0.
- Lemma 5: For a growth-restricted metric space (V,d) with expansion rate c, a (r,4,c3)-partition can be computed efficiently for any r>0.

UTSP detail-5

- Theorem 1: Given a metric space (V,d) with n vertices and a (,I)-partitioning scheme, UTSP-ALG returns a tour with stretch O(2I log n) in polynomial time.

UTSP detail-6

- Corollary 1: For any metric space over n vertices, the algorithm UTSP-ALG returns a tour with stretch O(log4(n)/log(log(n))).
- Corollary 2: For any doubling. Euclidean, or growth-restricted metric space over n vertices, UTSP-ALG return a tree with stretch O(log(n)).

UTSP detail-6

- UTSP-ALG runing in p time.
- For j 1, let PSj be a maxima subset of vertices S that are pairwise separated by distance at least j-1.
- Lemma 6: For j2, the cost of edges in Ts at level j is at most |PSj-1|jI.
- Lemma 7: If mi 1, i=1,…,k,c 2, then 1 i kmici (logc(max1 i k mi)+3) max1 i k (mi ci)

Other result-1

- Universal steiner trees
- Theorem 2: Given a metric space (V,d) with n vertices, a root rV, and a (,I)-partitioning scheme for (V,d), a spanning tree with stretch O(2I log(n)) can be constructed in polynomial time.
- Corollary 3: For any metric space over n vertices, UST-ALG returns a tree with stretch O(log4(n)/loglog(n)).
- Corollary 4: For any doubling, Euclidean, or growth-restricted metric space over n vertices, UST-ALG returns a tree with stretch O(log(n)).

Other result-2

- Theorem 3: There exists a set of n vertices in two-dimensional Euclidean space, for which every spanning tree has an (log(n)/loglog(n)) stretch.
- Theorem 4: No on-line algorithm can achieve a competitive ratio which is better than (log(n)/loglog(n)) for the Steiner tree problem of n vertices in vertices in the plane, or even for n vertices in the n by n grid.

Other result-3

- Theorem 5: For any USC instance with n elements, USC-ALG has a stretch of O((nln(n))1/2).
- Theorem 6: There exists an n-element instance of USC with uniform costs for which the best stretch achievable is ( n1/2).

Homework-2

- Try to find a natural problem and define it to be a universal approximation problem.

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