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Approximation algorithms for the TSP with a Parameterized Triangle InequalityRotem GolanAlon Furman

Department of Computer ScienceBen-Gurion University1Outline2The ProblemAn instance of the problem is a weighted graph G consisting on n vertices (representing cities).Distances between vertices satisfy the -inequality and the length of an optimal TSP tour is denoted OPT.

-inequality: dist(x,z) (dist(x,y) + dist(y,z))

The objective of the problem is to find (in polynomial time) a tour having length at most f()*OPT, where f()is a slowly-growing function of .33We observe two well known approximation algorithms for the TSP:Double-Tree Algorithm1. Find a minimum spanning tree T for G.2. Create a multigraph by duplicating each edge in the MST.3. Find an Eulerian tour.4. Shortcut over nodes to obtain a Hamiltonian cycle.

4The above algorithm yields a 2-approximation.

567Notice that step 4 in both algorithms obtains a Hamiltonian cycle by shortcutting the Eulerian tour.In the TSP (satisfying the triangle inequality) the above step will always result in a shorter path, however, this is not the case for the -TSP!

81015dist(x,y) (dist(x,z) + dist(z,y)) 2 (15+10) 50 = 2xy50z9Outline10TSP-ApproxDefinitions:A graph S is 2 node connected if the deletion of any node from S leaves it connected.

1112Observations:Any Hamiltonian cycle is 2-node connected. Thus, the minimum weight 2-node connected subgraph is a lower bound on OPT.It is NP hard to find the minimum weight 2-node connected subgraph. However, 2 approximation can be computed in polynomial time (Khuller & Raghavachari 1996, Penn & Shasha-Krupnik 1995).The square of 2-node connected graph contains a Hamilton cycle, which can be found in polynomial time (Fleischner 1970 ,Lau 1980).

13TSP-Approx Algorithm1415

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Theorem 1: Algorithm TSP-Approx is a 4 approximation algorithm.

We obtain a 4 approximation by finding a new Hamiltonian cycle C that crosses every edge in S at most twice.

We obtain C by adjusting overlapping edges, there are 2 cases.

17Case 1:The edge {u,v} is crossed 4 times.Hamiltonian cycle C can be modified so that the edge {u,v} is never crossed.

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Case 2:The edge {u,v} is crossed 3 times.Hamiltonian cycle C can be modified so that the edge {u,v} is crossed only once.19

ExampleFind an (approximately) minimum cost 2-node connected subgraph of G. Call this graph S.

2021ab(a,b) is being crossed 3 timesab(a,b) is being crossed 1 timeAdjust the overlapping edgesConclusions22Outline232425Conclusions26Outline2728The Algorithm29303132ExampleGiven graph G well find the minimum spanning tree T33343213353637383940=4142=H =43H =Outline446-approximation algorithm45Why Euclidean TSP does not work in this case?

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Conclusion52