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20
Asymmetric Traveling Salesman Path and Directed Latency Problems
, 2009
"... We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric d and two specified nodes s and t, both problems ask to find an st path visiting all other nodes. In the asymmetric traveling salesman path p ..."
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We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric d and two specified nodes s and t, both problems ask to find an st path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NPhard. The best known approximation algorithms for ATSPP have ratios O(log n) [5, 7], but only a bound of O ( √ n) for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n 1/2+ɛ), for any constant ɛ> 0 [19]. We present a new algorithm for the ATSPP problem that has approximation ratio of O(log n), matching the best known ones, but whose analysis also bounds the integrality gap of the LP relaxation of ATSPP by the same factor. This solves an open problem posed in [5]. We then pursue a deeper study of this LP and its variations and their use in approximating directed latency. Our second major result is an O(log n)approximation to the directed latency problem. This also places an O(log n) bound on the integrality gap of a new LP relaxation of the latency problem that we introduce. We note that this is essentially the best possible ratio unless an asymptotically better approximation exists for the wellstudied asymmetric traveling salesman tour problem. 1
A Dynamic Traveling Salesman Problem with Stochastic Arc Costs
 Operations Research
"... {toriello, wbhaskel, poremba} at usc dot edu ..."
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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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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.
A rounding by sampling approach to the minimum size karc connected subgraph problem
 In ICALP
, 2012
"... In the karc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least karc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)approximation to the unweighted variant of the prob ..."
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In the karc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least karc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. [GGTW09]. Similar to the 2approximation algorithm for this problem [FJ81], our algorithm simply takes the union of a k inarborescence and a k outarborescence. The main difference is in the selection of the two arborescences. Here, inspired by the recent applications of the rounding by sampling method (see e.g. [AGM+10, MOS11, OSS11, AKS12]), we select the arborescences randomly by sampling from a distribution on unions of k arborescences that is defined based on an extreme point solution of the linear programming relaxation of the problem. In the analysis, we crucially utilize the sparsity property of the extreme point solution to upperbound the size of the union of the sampled arborescences. To complement the algorithm, we also show that the integrality gap of the minimum cost strongly connected subgraph problem (i.e., when k = 1) is at least 3/2 − , for any > 0. Our integrality gap instance is inspired by the integrality gap example of the asymmetric traveling salesman problem [CGK06], hence providing further evidence of connections between the approximability of the two problems. 1
Equivalence of an Approximate Linear Programming Bound with the HeldKarp Bound for the Traveling Salesman Problem
"... atoriello at isye dot gatech dot edu ..."
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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
Recent Advances in Approximation Algorithms Spring 2015 Lecture 17: Cheeger’s Inequality and the Sparsest Cut Problem
"... Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In the rest of this course we use the proof of Marcus, Spielman and Srivastava to prove an upper bound of polyloglog(n) on the integrality gap of the HeldKarp relaxation for ATSP. The materials w ..."
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Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In the rest of this course we use the proof of Marcus, Spielman and Srivastava to prove an upper bound of polyloglog(n) on the integrality gap of the HeldKarp relaxation for ATSP. The materials will be based on the work of Oveis Gharan and Anari [AO14]. We start by introducing the sparsest cut problem and Cheeger’s inequalities. The ideas that we develop here will be crucially used later. In particular, we will see that the ideas involved in the semidefinite programming relaxation of the sparsest cut can be used to design an algorithm for finding thin trees. For a graph G = (V,E), the conductance of a set S is the ratio of the fraction of edges in the cut (S, S) to the volume of S, φ(S):= E(S, S) vol(S) where vol(S) = v∈S d(v) is the sum of the degree of vertices in S. Observe that for any set S ⊆ V, 0 ≤ φ(S) ≤ 1. If φ(S) ≈ 0, S may represent a cluster in G. Conductance is a very well studied measure for graph clustering in the literature (see e.g. [SM00; KVV04; TM06]). The conductance of G, φ(G) is the smallest conductance among all sets with at most half of the total volume, φ(G) = min S:vol(S)≤vol(V)/2 φ(S). For example, the conductance of a complete graph Kn is φ(Kn) ≈ 1/2, and the worst set is any set with half of the vertices. The conductance of a cycle (of length n), Cn is about 2/n, φ(Cn) ≈ 2/n and the worst set is a path of length n/2. We say a graph G is an expander if φ(G) ≥ Ω(1). In this lecture we will see several properties of expander graphs. Throughout this lecture we assume that G is a dregular unweighted graph but the statements naturally extend to weighted nonregular graphs. 17.1 Cheeger’s inequality Cheeger’s inequality is perhaps one of the most fundamental inequalities in Discrete optimization, spectral graph theory and the analysis of Markov Chains. It relates the eigenvalue of the normalized Laplacian matrix to φ(G). It has many applications in graph clustering [ST96; KVV04], explicit construction of expander graphs [JM85; HLW06; Lee12], analysis of Markov chains [SJ89; JSV04], and image segmentation [SM00]. The normalized Laplacian matrix of G, LG is defined as follows
Optimal MultiMeetingPoint Route Search
"... Abstract—Realtime ridesharing applications (e.g., Uber and Lyft) are very popular in recent years. Motivated by the ridesharing application, we propose a new type of query in road networks, called the optimal multimeetingpoint route (OMMPR) query. Given a road network G, a source node s, a targ ..."
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Abstract—Realtime ridesharing applications (e.g., Uber and Lyft) are very popular in recent years. Motivated by the ridesharing application, we propose a new type of query in road networks, called the optimal multimeetingpoint route (OMMPR) query. Given a road network G, a source node s, a target node t, and a set of query nodes U, the OMMPR query aims at finding the best route starting from s and ending at t such that the weighted average cost between the cost of the route and the total cost of the shortest paths from every query node to the route is minimized. We show that the problem of computing the OMMPR query is NPhard. To answer the OMMPR query efficiently, we propose two novel parameterized solutions based on dynamic programming (DP), with the number of query nodes l (i.e., l = jU j) as a parameter, which is typically very small in practice. The two proposed parameterized algorithms run in O(3l m + 2l n (l + log(n))) and O(2l (m + n (l + log(n)))) time respectively, where n and m denote the number of nodes and edges in graph G, thus they are tractable in practice. To reduce the search space of the DPbased algorithms, we propose two novel optimized algorithms based on bidirectional DP and a carefullydesigned lower bounding technique. We conduct extensive experimental studies on four large realworld road networks, and the results demonstrate the efficiency of the proposed algorithms.