Zhihan Gao Department of Combinatorics and Optimization University - PowerPoint PPT Presentation

An LP-based 3 2 -approximation algorithm for the graphic s - t path TSP Zhihan Gao Department of Combinatorics and Optimization University of Waterloo June 11, 2013 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 1 / 13

Traveling Salesman Problem(TSP) TSP Given a complete graph with edge-cost w , TSP is to find a min-cost Hamiltonian cycle. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 2 / 13

Traveling Salesman Problem(TSP) TSP Given a complete graph with edge-cost w , TSP is to find a min-cost Hamiltonian cycle. For arbitrary cost w , TSP is NP-hard to approximate. If we restrict w to be metric cost, it is called metric TSP , i.e., w satisfies the triangle inequality. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 2 / 13

Traveling Salesman Problem(TSP) TSP Given a complete graph with edge-cost w , TSP is to find a min-cost Hamiltonian cycle. For arbitrary cost w , TSP is NP-hard to approximate. If we restrict w to be metric cost, it is called metric TSP , i.e., w satisfies the triangle inequality. Metric TSP Given a connected graph G with edge cost c , metric TSP is to find a min-cost Hamiltonian cycle in the metric completion of G . LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 2 / 13

s-t Path TSP Path version of metric TSP Let G be a connected graph with edge cost c . Given two vertices s , t ∈ V ( G ) , s-t Path TSP is to find a min-cost Hamiltonian path from s to t in the metric completion of G . If we restrict c to be a unit cost, it is called graphic s-t path TSP . LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 3 / 13

Literature Metric s-t path TSP Due to Year Approx. Factor 5 J. A. Hoogeveen 1991 3 ≈ 1 . 666 √ 1 + 5 ≈ 1 . 618 H-C An et al (AKS) 2012 2 8 A. Seb˝ o 2012 5 = 1 . 6 Graphic s-t path TSP Due to Year Approx. Factor ( 5 H-C An & D. Shmoys 2011 3 − ǫ ) T. M¨ omke & O.Svensson 2011 1 . 586 19 12 + ǫ ≈ 1 . 58333 + ǫ M. Mucha 2012 A. Seb˝ o & J. Vygen 2012 1 . 5 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 4 / 13

Equivalent Definition of graphic s-t Path TSP Graphic s-t Path TSP G (Original Graph) : Connected graph with unit edge-cost c . s-t path TSP: For s , t ∈ V ( G ) , find a minimum cost Hamiltonian path from s to t in the metric completion of G . Equivalent definition on original graph: Find a minimum-size connected spanning subgraph of 2 G with { s , t } as the odd-degree vertex set . LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 5 / 13

Scheme of Algorithm Find a spanning tree J of G . 1 —– Build Connectivity Find min-size D -join F for the spanning tree J . 2 D : wrong degree vertex set of J D -join: subgraph with D as odd-degree vertex set —– Correct Degree Output J ˙ ∪ F . 3 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 6 / 13

Scheme of Algorithm Find a spanning tree J of G . 1 —– Build Connectivity Find min-size D -join F for the spanning tree J . 2 D : wrong degree vertex set of J D -join: subgraph with D as odd-degree vertex set —– Correct Degree Output J ˙ ∪ F . 3 Question How to bound the min-size of D -join ? How to find a ’good’ spanning tree for the algorithm? LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 6 / 13

LP relaxations s - t Path TSP (on 2 G ): x ∗ optimal solution minimize : � e ∈ E x e x ( δ ( W )) ≥ |W| − 1 ∀ partition W of V subject to : x ( δ ( S )) ≥ 2 ∀∅ � S � V , | S ∩ { s , t }| even 2 ≥ x e ≥ 0 ∀ e ∈ E LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 7 / 13

LP relaxations s - t Path TSP (on 2 G ): x ∗ optimal solution minimize : � e ∈ E x e x ( δ ( W )) ≥ |W| − 1 ∀ partition W of V subject to : x ( δ ( S )) ≥ 2 ∀∅ � S � V , | S ∩ { s , t }| even 2 ≥ x e ≥ 0 ∀ e ∈ E D -join (Wrong-degree vertex set D for the spanning tree J ) : minimize : � e ∈ E x e subject to : x ( δ ( S )) ≥ 1 ∀ D -odd S ≥ ∀ e ∈ E x e 0 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 7 / 13

LP relaxations s - t Path TSP (on 2 G ): x ∗ optimal solution minimize : � e ∈ E x e x ( δ ( W )) ≥ |W| − 1 ∀ partition W of V subject to : x ( δ ( S )) ≥ 2 ∀∅ � S � V , | S ∩ { s , t }| even 2 ≥ x e ≥ 0 ∀ e ∈ E D -join (Wrong-degree vertex set D for the spanning tree J ) : minimize : � e ∈ E x e subject to : x ( δ ( S )) ≥ 1 ∀ D -odd S ≥ ∀ e ∈ E x e 0 Bad Cuts S : x ∗ ( S ) < 2, D − odd and aslo s , t -odd. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 7 / 13

LP relaxations s - t Path TSP (on 2 G ): x ∗ optimal solution minimize : � e ∈ E x e x ( δ ( W )) ≥ |W| − 1 ∀ partition W of V subject to : x ( δ ( S )) ≥ 2 ∀∅ � S � V , | S ∩ { s , t }| even 2 ≥ x e ≥ 0 ∀ e ∈ E D -join (Wrong-degree vertex set D for the spanning tree J ) : minimize : � e ∈ E x e subject to : x ( δ ( S )) ≥ 1 ∀ D -odd S ≥ ∀ e ∈ E x e 0 Bad Cuts S : x ∗ ( S ) < 2, D − odd and aslo s , t -odd. 2 x ∗ fixes our spanning tree. If no bad cuts, 1 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 7 / 13

Idea: No Bad Cuts Lemma For a spanning J with wrong degree vertex set D , if a cut S is both D − odd and s , t -odd, then | J ∩ δ ( S ) | is EVEN . Idea: Construct a spanning tree that has ODD number of edges at the potential Bad Cuts. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 8 / 13

Narrow Cuts: Potential Bad Cuts Bad Cuts S : x ∗ ( S ) < 2, D − odd and s , t -odd. Narrow cut S : x ∗ ( S ) < 2 and s , t -odd cut. Bad cut must be a narrow cut. Construct a spanning tree that has ODD number of edges at the narrow cuts. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 9 / 13

Narrow Cuts: Potential Bad Cuts Bad Cuts S : x ∗ ( S ) < 2, D − odd and s , t -odd. Narrow cut S : x ∗ ( S ) < 2 and s , t -odd cut. Bad cut must be a narrow cut. Construct a spanning tree that has ODD number of edges at the narrow cuts. AKS’s Lemma on Narrow Cuts Narrow cuts is a nested family. s L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 t LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 9 / 13

Key Lemma Lemma Let H be the support graph of an optimal solution x ∗ of s-t path TSP LP . Then, H ( ∪ p ≤ i ≤ q L i ) is connected. . . . s L p L q t H ( ∪ p ≤ i ≤ q L i ) connected. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 10 / 13

Key Lemma Lemma Let H be the support graph of an optimal solution x ∗ of s-t path TSP LP . Then, H ( ∪ p ≤ i ≤ q L i ) is connected. H ( L i ) is connected, and ∃ edge connecting L i and L i + 1 in H . . . . s L p L q t H ( ∪ p ≤ i ≤ q L i ) connected. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 10 / 13

Key Lemma Lemma Let H be the support graph of an optimal solution x ∗ of s-t path TSP LP . Then, H ( ∪ p ≤ i ≤ q L i ) is connected. H ( L i ) is connected, and ∃ edge connecting L i and L i + 1 in H . . . . s L p L q t H ( ∪ p ≤ i ≤ q L i ) connected. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 10 / 13

Key Lemma Lemma Let H be the support graph of an optimal solution x ∗ of s-t path TSP LP . Then, H ( ∪ p ≤ i ≤ q L i ) is connected. H ( L i ) is connected, and ∃ edge connecting L i and L i + 1 in H . . . . s L p L q t H ( ∪ p ≤ i ≤ q L i ) connected. LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 10 / 13

LP-based Algorithm LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 11 / 13

LP-based Algorithm Find the support graph H of x ∗ . 1 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 11 / 13

LP-based Algorithm Find the support graph H of x ∗ . 1 Find all narrow cuts and corresponding partition L 1 , L 2 , . . . , L k + 1 . 2 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 11 / 13

LP-based Algorithm Find the support graph H of x ∗ . 1 Find all narrow cuts and corresponding partition L 1 , L 2 , . . . , L k + 1 . 2 For each 1 ≤ i ≤ k + 1, find a local spanning tree J i on H ( L i ) . 3 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 11 / 13

LP-based Algorithm Find the support graph H of x ∗ . 1 Find all narrow cuts and corresponding partition L 1 , L 2 , . . . , L k + 1 . 2 For each 1 ≤ i ≤ k + 1, find a local spanning tree J i on H ( L i ) . 3 For each 1 ≤ i ≤ k , take an edge e i connecting each two 4 consecutive L i and L i + 1 LP-based 3 Zhihan Gao (University of Waterloo) 2 -approximation June 11, 2013 11 / 13