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22
Improved Inapproximability for TSP
, 2014
"... The Traveling Salesman Problem is one of the most studied problems in the theory of algorithms and its approximability is a longstanding open question. Prior to the present work, the best known inapproximability threshold was 220/219, due to Papadimitriou and Vempala. Here, using an essentially di ..."
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The Traveling Salesman Problem is one of the most studied problems in the theory of algorithms and its approximability is a longstanding open question. Prior to the present work, the best known inapproximability threshold was 220/219, due to Papadimitriou and Vempala. Here, using an essentially different construction and also relying on the work of Berman and Karpinski on boundedoccurrence CSPs, we give an alternative and simpler inapproximability proof which improves the bound to 185/184.
New Inapproximability Bounds for TSP
, 2015
"... In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric c ..."
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In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to Papadimitriou and Vempala). We construct here two new bounded occurrence CSP reductions which improve these bounds to 123/122 and 75/74, respectively. The latter bound is the first improvement in more than a decade for the case of the asymmetric TSP. One of our main tools, which may be of independent interest, is a new construction of a bounded degree wheel amplifier used in the proof of our results.
LPbased approximation algorithms for traveling salesman path problems. arXiv manuscript number 1105.2391
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EightFifth Approximation for TSP Paths
, 2012
"... We prove the approximation ratio 8/5 for the metric {s, t}pathTSP problem, and more generally for shortest connected Tjoins. The algorithm that achieves this ratio is the simple “Best of Many ” version of Christofides’ algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists ..."
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Cited by 3 (0 self)
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We prove the approximation ratio 8/5 for the metric {s, t}pathTSP problem, and more generally for shortest connected Tjoins. The algorithm that achieves this ratio is the simple “Best of Many ” version of Christofides’ algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides {s, t}tour out of those constructed from a family F>0 of trees having a convex combination dominated by an optimal solution x ∗ of the fractional relaxation. They give the approximation guarantee 5+1 2 for such an {s, t}tour, which is the first improvement after the 5/3 guarantee of Hoogeveen’s Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8approximation of shortest connected Tjoins, for T  ≥ 4. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x∗/2 in order to dominate the cost of “parity correction” for spanning trees. We partition the edgeset of each spanning tree in F>0 into an {s, t}path (or more generally, into a Tjoin) and its complement, which induces a decomposition of x∗. This decomposition can be refined and then efficiently used to complete x∗/2 without using linear programming or particular properties of T, but by adding to each cut deficient for x∗/2 an individually tailored explicitly given vector, inherent in x∗. A simple example shows that the Best of Many Christofides algorithm may not find a shorter {s, t}tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.
TSP tours in cubic graphs: Beyond 4/3
, 2015
"... After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65–77] proved that any 2connected graph whose n vertices have degree 3, i.e., a cubic 2connected ..."
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After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65–77] proved that any 2connected graph whose n vertices have degree 3, i.e., a cubic 2connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3−1/61236)n, implying that cubic 2connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller , we show that the integrality gap of the TSP relaxation is at most 4/3 − even if the graph is not 2connected (i.e., for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 − 1/18)n.
Improved approximation algorithms for MinMax Tree Cover, Bounded Tree Cover, ShallowLight and BuyatBulk kSteiner Tree, and (k, 2)Subgraph
, 2011
"... In this thesis we provide improved approximation algorithms for the MinMax kTree Cover, Bounded Tree Cover and ShallowLight kSteiner Tree, (k, 2)subgraph problems. In Chapter 2 we consider the MinMax kTree Cover (MMkTC). Given a graph G = (V, E) with weights w: E → Z +, a set T1, T2,..., Tk ..."
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In this thesis we provide improved approximation algorithms for the MinMax kTree Cover, Bounded Tree Cover and ShallowLight kSteiner Tree, (k, 2)subgraph problems. In Chapter 2 we consider the MinMax kTree Cover (MMkTC). Given a graph G = (V, E) with weights w: E → Z +, a set T1, T2,..., Tk of subtrees of G is called a tree cover of G if V = ⋃ k i=1 V (Ti). In the MMkTC problem we are given graph G and a positive integer
Approximation Hardness of Graphic TSP on Cubic Graphs
, 2013
"... We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcub ..."
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We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used in the paper could be also of independent interest.
LiftandProject Integrality Gaps for the Traveling Salesperson Problem
, 2011
"... We study the liftandproject procedures of LovászSchrijver and SheraliAdams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integral ..."
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We study the liftandproject procedures of LovászSchrijver and SheraliAdams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integrality gap of the standard relaxation is at least 2. We prove that after one round of the LovászSchrijver or SheraliAdams procedures, the integrality gap of the asymmetric TSP tour problem is at least 3/2, with a small caveat on which version of the standard relaxation is used. For the symmetric TSP tour problem, the integrality gap of the standard relaxation is known to be at least 4/3, and Cheung (SIOPT 2005) proved that it remains at least 4/3 after o(n) rounds of the LovászSchrijver procedure, where n is the number of nodes. For the symmetric TSP path problem, the integrality gap of the standard relaxation is known to be at least 3/2, and we prove that it remains at least 3/2 after o(n) rounds of the LovászSchrijver procedure, by a simple reduction to Cheung’s result. 1
Short Tours through Large Linear Forests
"... Abstract. A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected dregular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His pr ..."
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Abstract. A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected dregular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on vanderWarden’s conjecture (proved independently by Egorychev (1981) and by Falikman (1981)) regarding the permanent of doubly stochastic matrices. We provide an exponential improvement in the rate of decrease of the o(1) term (thus increasing the range of d for which the upper bound on the tour length is nontrivial). Our proof does not use the vanderWarden conjecture, and instead is related to the linear arboricity conjecture of Akiyama, Exoo and Harary (1981), or alternatively, to a conjecture of Magnant and Martin (2009) regarding the path cover number of regular graphs. More generally, for arbitrary connected graphs, our techniques provide an upper bound on the minimum tour length, expressed as a function of their maximum, average, and minimum degrees. Our bound is best possible up to a term that tends to 0 as the minimum degree grows. 1