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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
On the integrality gap of the subtour LP for the 1,2TSP
 In Proc. LATIN 2012, volume 7256 of Lecture Notes Comput. Sci
, 2012
"... Abstract. In this paper, we study the integrality gap of the subtour LP relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2. For the general case of symmetric costs that obey triangle inequality, a famous conjecture is that the integrality gap is 4 ..."
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Abstract. In this paper, we study the integrality gap of the subtour LP relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2. For the general case of symmetric costs that obey triangle inequality, a famous conjecture is that the integrality gap is 4/3. Little progress towards resolving this conjecture has been made in thirty years. We conjecture that when all edge costs cij ∈ {1, 2}, the integrality gap is 10/9. We show that this conjecture is true when the optimal subtour LP solution has a certain structure. Under a weaker assumption, which is an analog of a recent conjecture by Schalekamp, Williamson and van Zuylen, we show that the integrality gap is at most 7/6. When we do not make any assumptions on the structure of the optimal subtour LP solution, we can show that inegrality gap is at most 19/15 ≈ 1.267 < 4/3; this is the first bound on the integrality gap of the subtour LP strictly less than 4/3 known for an interesting special case of the TSP. 1
GraphTSP from Steiner Cycles
"... Abstract. We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph G, if we can find a spanning tree T and a simple cycle that contains the vertices with ..."
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Abstract. We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph G, if we can find a spanning tree T and a simple cycle that contains the vertices with odddegree in T, then we show how to combine the classic “double spanning tree ” algorithm with Christofides ’ algorithm to obtain a TSP tour of length at most 4n 3. We use this approach to show that a graph containing a Hamiltonian path has a TSP tour of length at most 4n/3. Since a Hamiltonian path is a spanning tree with two leaves, this motivates the question of whether or not a graph containing a spanning tree with few leaves has a short TSP tour. The recent techniques of Mömke and Svensson imply that a graph containing a depthfirstsearch tree with k leaves has a TSP tour of length 4n/3+O(k). Using our approach, we can show that a 2(k − 1)vertex connected graph that contains a spanning tree with at most k leaves has a TSP tour of length 4n/3. We also explore other conditions under which our approach results in a short tour. 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
A Note on Using Tjoins to Approximate Solutions for min Graphic kPath TSP
, 2012
"... We consider a generalized Path Traveling Salesman Problem where the distances are defined by a 2edgeconnected graph metric and a constant number of salesmen have to cover all the destinations by traveling along paths of minimum total length. We show that for this problem there is a polynomial algo ..."
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We consider a generalized Path Traveling Salesman Problem where the distances are defined by a 2edgeconnected graph metric and a constant number of salesmen have to cover all the destinations by traveling along paths of minimum total length. We show that for this problem there is a polynomial algorithm with asymptotic approximation ratio of 3/2.
Approximation algorithms and the hardness of approximation
, 2011
"... Most of the many discrete optimization problems arising in the sciences, engineering, and mathematics are NPhard, that is, there exist no efficient algorithms to solve them to optimality, assuming the P=NP conjecture. The area of approximation algorithms focuses on the design and analysis of effici ..."
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Most of the many discrete optimization problems arising in the sciences, engineering, and mathematics are NPhard, that is, there exist no efficient algorithms to solve them to optimality, assuming the P=NP conjecture. The area of approximation algorithms focuses on the design and analysis of efficient algorithms that find solutions that are within a guaranteed factor of the optimal one. Loosely speaking, in the context of studying algorithmic problems, an approximation guarantee captures the “goodness ” of an algorithm – for every possible set of input data for the problem, the algorithm finds a solution whose cost is within this factor of the optimal cost. A hardness threshold indicates the “badness ” of the algorithmic problem – no efficient algorithm can achieve an approximation guarantee better than the hardness threshold assuming that P=NP (or a similar complexity assumption). Over the last two decades, there have been major advances on the design and analysis of approximation algorithms, and on the complementary topic of the hardness of approximation, see [33], [34]. The longterm agenda of our area (approximation algorithms and hardness results) is to classify all of the fundamental NPhard problems according to their approximability and hardness thresholds. This agenda may seem farfetched, but remarkable progress has been made over the last two decades. Approximation guarantees and hardness thresholds that “match ” each other have been established for key problems in topics
Recall: LinearTime Approximation Schemes for Planar Graphs (L. 8) Example min VERTEXCOVER
, 2011
"... Traveling Salesman Problem (TSP) given G = (V,E) find a tour visiting each1 node v ∈ V. NP–hard optimization problem, hard even for planar graphs Polynomialtime approximation for general graphs: Christofides ’ algorithm achieves 3/2 approximation Assumption (all of Lecture 15) undirected planar G, ..."
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Traveling Salesman Problem (TSP) given G = (V,E) find a tour visiting each1 node v ∈ V. NP–hard optimization problem, hard even for planar graphs Polynomialtime approximation for general graphs: Christofides ’ algorithm achieves 3/2 approximation Assumption (all of Lecture 15) undirected planar G, ` : E → R+ 2–approximation simple algorithm, bound approximation ratio in terms of minimum spanning tree • compute minimum spanning tree T. let `(T): = ∑ e∈T `(e) • duplicate all edges Eulerian graph • find Eulerian cycle, length at most 2`(T) • (if G is the complete graph Kn, Eulerian cycle can be converted into Hamiltonian cycle by skipping already visited nodes) any tour needs to visit all nodes, total length at least `(T), hence 2–approximation
APPROXIMATION ALGORITHMS FOR SOME MINMAX VEHICLE ROUTING PROBLEMS
, 2013
"... Vehicle routing problems are optimization problems that deal with the location and routing of vehicles. A set of clients based in different locations need to be served by a fleet of vehicles. The clients and vehicle depots are modeled as being placed on the vertices of a graph and the distances b ..."
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Vehicle routing problems are optimization problems that deal with the location and routing of vehicles. A set of clients based in different locations need to be served by a fleet of vehicles. The clients and vehicle depots are modeled as being placed on the vertices of a graph and the distances between them as a metric. Thus, a solution to a vehicle routing problem corresponds to covering the graph using a number of subgraphs, each denoting the route of a vehicle. In this thesis, we consider minmax vehicle routing problems, in which the maximum cost incurred by the subgraph corresponding to each vehicle is to be minimized. We study two types of covering problems and present new or improved approximation algorithms for them.