We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time 9 7 -approximation algorithm for cubic bipartite graphs and a ( 9 7 + 1 21(k-2) )-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of the structure of cubic graphs.
One value related to the ability to approximate TSP is the integrality gap, which is the worst-case ratio between the optimal solution for a TSP instance and the solution to a linear programming relaxation called the subtour relaxation @cite_1 . A long-standing conjecture (see, e.g., @cite_5 ) for Metric TSP is that the integrality gap is @math . One source of motivation for studying Graphic TSP is that the family of graphs with two vertices connected by three paths of length @math has an integrality gap that approaches @math . This family of graphs demonstrates that Graphic TSP captures much of the complexity of the more general Metric TSP problem.
@cite_5 @cite_1
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We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4 3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified. We present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5 3-approximation algorithm for this problem, and this asymptotically tight bound in fact had been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5 2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP. We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7 5. For a generalization, the connected- @math -join problem, we obtain the first nontrivial approximation algorithm, with ratio 3 2. This contains the graphic @math - @math -path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4 3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios. We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3 2-approximation algorithm for TSP in general. We consider the metric Traveling Salesman Problem (Δ-TSP for short) and study how stability (as defined by Bilu and Linial [3]) influences the complexity of the problem. On an intuitive level, an instance of Δ-TSP is γ-stable (γ > 1), if there is a unique optimum Hamiltonian tour and any perturbation of arbitrary edge weights by at most γ does not change the edge set of the optimal solution (i.e., there is a significant gap between the optimum tour and all other tours). We show that for γ ≥ 1.8 a simple greedy algorithm (resembling Prim's algorithm for constructing a minimum spanning tree) computes the optimum Hamiltonian tour for every γ-stable instance of the Δ-TSP, whereas a simple local search algorithm can fail to find the optimum even if γ is arbitrary. We further show that there are γ-stable instances of Δ-TSP for every 1 < γ < 2. These results provide a different view on the hardness of the Δ-TSP and give rise to a new class of problem instances which are substantially easier to solve than instances of the general Δ-TSP. We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios. The traveling salesman problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem an algorithm that for any fixed @math computes in randomized polynomial time a @math -approximation to the optimal tour in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [J. ACM, 45 (1998), pp. 753--782] and Mitchell [SIAM J. Comput., 28 (1999), pp. 1298--1309] prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 281--290]. The Held-Karp heuristic for the Traveling Salesman Problem (TSP) has in practice provided near-optimal lower bounds on the cost of solutions to the TSP. We analyze the structure of Held-Karp solutions in order to shed light on their quality. In the symmetric case with triangle inequality, we show that a class of instances has planar solutions. We also show that Held-Karp solutions have a certain monotonicity property. This leads to an alternate proof of a result of Wolsey, which shows that the value of Held-Karp heuristic is always at least 2 3 OPT, where OPT is the cost of the optimum TSP tour. Additionally, we show that the value of the Held-Karp heuristic is equal to that of the linear relaxation of the biconnected-graph problem when edge costs are non-negative. In the asymmetric case with triangle inequality, we show that there are many equivalent definitions of the Held-Karp heuristic, which include finding optimally weighted 1-arborescences, 1-antiarborescences, asymmetric 1-trees, and assignment problems. We prove that monotonicity holds in the asymmetric case as well. These theorems imply that the value of the Held-Karp heuristic is no less than OPT and no less than the value of the Balas-Christofides heuristic for the asymmetric TSP. For the 1,2-TSP, we show that the Held-Karp heuristic cannot do any better than 9 10 OPT, even as the number of nodes tends to infinity. Portions of this thesis are joint work with David Shmoys. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. 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 @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math 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 8-approximation of shortest connected T-joins, 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 edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) 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. Abstract In their 1971 paper on the travelling salesman problem and minimum spanning trees, Held and Karp showed that finding an optimally weighted 1-tree is equivalent to solving a linear program for the traveling salesman problem (TSP) with only node-degree constraints and subtour elimination constraints. In this paper we show that the Held-Karp 1-trees have a certain monotonicity property: given a particular instance of the symmetric TSP with triangle inequality, the cost of the minimum weighted 1-tree is monotonic with respect to the set of nodes included. As a consequence, we obtain an alternate proof of a result of Wolsey and show that linear programs with node-degree and subtour elimination constraints must have a cost at least 2 3 OPT where OPT is the cost of the optimum solution to the TSP instance. Abstract : An O(n sup 3) heuristic algorithm is described for solving n-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition. The algorithm involves as substeps the computation of a shortest spanning tree of the graph G defining the TSP, and the finding of a minimum cost perfect matching of a certain induced subgraph of G. A worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3 2. This represents a 50 reduction over the value 2 which was the previously best known such ratio for the performance of other polynomial-growth algorithms for the TSP. Constraint Programming (CP) has been successfully applied to several combinatorial optimization problems. One of its advantages is the availability of complex global constraints performing efficient propagation and interacting with each other through shared variables. However, CP techniques have shown their limitations in dealing with optimization problems since the link between the objective function and problem decision variables is often quite loose and does not produce an effective propagation. We propose to integrate optimization components in global constraints, aimed at optimally solving a relaxation corresponding to the constraint itself. The optimal solution of the relaxation provides pieces of information which can be exploited in order to perform pruning on the basis of cost-based reasoning. In fact, we exploit reduction rules based on lower bound and reduced costs calculation to remove those branches which cannot improve the best solution found so far. The interest of integrating efficient well-known Operations Research (OR) algorithms into CP is mainly due to the smooth interaction between CP domain reduction and information provided by the relaxation acting on variable domains which can be seen as a i>communication channel among different techniques. We have applied this technique to symmetric and asymmetric Traveling Salesman Problem (TSP) instances both because the TSP is an interesting problem arising in many real-life applications, and because pure CP techniques lead to disappointing results for this problem. We have tested the proposed optimization constraints using ILOG solver. Computational results on benchmarks available from literature, and comparison with related approaches are described in the paper. The proposed method on pure TSPs improves the performances of CP solvers, but is still far from the OR state of the art techniques for solving the problem. However, due to the flexibility of the CP framework, we could easily use the same technique on TSP with Time Windows, a time constrained variant of the TSP. For this type of problem, we achieve results that are comparable with state of the art OR results. The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+µ)-approximation to the optimal tour, for any fixed µ>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [Aro98] and Mitchell [Mit99] prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar [Tal04]. In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of @math and a randomized approximation algorithm that achieves a ratio of @math . In particular, we obtain a 2+e approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio @math ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio @math ), STSP with weights one and two (ratio 4 3) and ATSP with weights one and two (ratio 3 2). The traveling salesman problem (TSP) is a canonical NP-complete problem which is proved by Trevisan [SIAM J. Comput., 30 (2000), pp. 475--485] to be MAX-SNP hard even on high-dimensional Euclidean metrics. To circumvent this hardness, researchers have been developing approximation schemes for „simpler” instances of the problem. For instance, the algorithms of Arora and of Talwar show how to approximate TSP on low-dimensional metrics (for different notions of metric dimension). However, a feature of most current notions of metric dimension is that they are „local”: the definitions require every local neighborhood to be well-behaved. In this paper, we define a global notion of dimension that generalizes the popular notion of doubling dimension, but still allows some small dense regions; e.g., it allows some metrics that contain cliques of size @math . Given a metric with global dimension @math , we give a @math -approximation algorithm that runs in subexponential time, i.e., in $ (O(n^... We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem. Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound the runtime of simple evolutionary algorithms. Our analysis studies the runtime in dependence of the number of inner points @math and shows that @math evolutionary algorithms solve the Euclidean TSP in expected time @math where @math is a function of the minimum angle @math between any three points. Finally, our analysis provides insights into designing a mutation operator that improves the upper bound on expected runtime. We show that a mixed mutation strategy that incorporates both 2-opt moves and permutation jumps results in an upper bound of @math for the @math EA. We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7 8-approximation for max-sat. We present randomized approximation algorithms for multi-criteria Max-TSP. For Max-STSP with k > 1 objective functions, we obtain an approximation ratio of @math for arbitrarily small @math . For Max-ATSP with k objective functions, we obtain an approximation ratio of @math .