A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints

This article presents an exact algorithm for the multi-depot vehicle routing problem (MDVRP) under capacity and route length constraints. The MDVRP is formulated using a vehicle-flow and a set-partitioning formulation, both of which are exploited at different stages of the algorithm. The lower bound computed with the vehicle-flow formulation is used to eliminate non-promising edges, thus reducing the complexity of the pricing subproblem used to solve the set-partitioning formulation. Several classes of valid inequalities are added to strengthen both formulations, including a new family of valid inequalities used to forbid cycles of an arbitrary length. To validate our approach, we also consider the capacitated vehicle routing problem (CVRP) as a particular case of the MDVRP, and conduct extensive computational experiments on several instances from the literature to show its effectiveness. The computational results show that the proposed algorithm is competitive against stateof-the-art methods for these two classes of vehicle routing problems, and is able to solve to optimality some previously open instances. Moreover, for the instances that cannot be solved by the proposed algorithm, the final lower bounds prove stronger than those obtained by earlier methods

Complexity of union-split-find problems

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 45-46).In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an (log n/ log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data structure that achieves a matching upper bound of (log n/ log log n) for Cyclic Union-Split Find in the word RAM model. For General Union-Split-Find, no o(n) bound is known. We present a data structure which has an [Omega](log2 n) amortized lower bound in the worst case that we conjecture has polylogarithmic amortized performance. This thesis is the product of joint work with Erik Demaine.by Katherine Jane Lai.M.Eng

Spectral Partitioning for Node Criticality

Finding critical nodes in a network is a significant task, highly relevant to network vulnerability and security. We consider the node criticality problem as an algebraic connectivity minimization problem where the objective is to choose nodes which minimize the algebraic connectivity of the resulting network. Previous suboptimal solutions of the problem suffer from the computational complexity associated with the implementation of a maximization consensus algorithm. In this work, we use spectral partitioning concepts introduced by Fiedler, to propose a new suboptimal solution which significantly reduces the implementation complexity. Our approach, combined with recently proposed distributed Fiedler vector calculation algorithms enable each node to decide by itself whether it is a critical node. If a single node is required then the maximization algorithm is applied on a restricted set of nodes within the network. We derive a lower bound for the achievable algebraic connectivity when nodes are removed from the network and we show through simulations that our approach leads to algebraic connectivity values close to this lower bound. Similar behaviour is exhibited by other approaches at the expense, however, of a higher implementation complexity

Approximating the Expansion Profile and Almost Optimal Local Graph Clustering

Spectral partitioning is a simple, nearly-linear time, algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. Local graph partitioning algorithms [ST08,ACL06,AP09] run in time that is nearly linear in the size of the output set, and their approximation guarantee is worse than the guarantee provided by the Cheeger inequalities by a polylogarithmic $\log^{\Omega(1)} n$ factor. It has been a long standing open problem to design a local graph clustering algorithm with an approximation guarantee close to the guarantee of the Cheeger inequalities and with a running time nearly linear in the size of the output. In this paper we solve this problem; we design an algorithm with the same guarantee (up to a constant factor) as the Cheeger inequality, that runs in time slightly super linear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger's inequality. As a byproduct of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. Let $\phi(\gamma) = \min_{\mu(S) \leq \gamma}\phi(S)$. There is a polynomial time algorithm that, for any $\gamma,\epsilon>0$, finds a set $S$ of measure $\mu(S)\leq 2\gamma^{1+\epsilon}$, and expansion $\phi(S)\leq \sqrt{2\phi(\gamma)/\epsilon}$. Our proof techniques also provide a simpler proof of the structural result of Arora, Barak, Steurer [ABS10], that can be applied to irregular graphs. Our main technical tool is that for any set $S$ of vertices of a graph, a lazy $t$-step random walk started from a randomly chosen vertex of $S$, will remain entirely inside $S$ with probability at least $(1-\phi(S)/2)^t$. This itself provides a new lower bound to the uniform mixing time of any finite states reversible markov chain

空間Webデータにおけるm-最近接キーワード検索問題のトップダウン解法に関する研究

This thesis addresses the problem of m-closest keywords queries (mCK queries) over spatial web objects that contain descriptive texts and spatial information. The mCK query is a problem to find the optimal set of records in the sense that they are the spatially-closest records that satisfy m user-given keywords in their texts. The mCK query can be widely used in various applications to find the place of user’s interest.　Generally, top-down search techniques using tree-style data structures are appropriate for finding optimal results of queries over spatial datasets. Thus in order to solve the mCK query problem, a previous study of NUS group assumed a specialized R*-tree (called bR*-tree) to store all records and proposed a top-down approach which uses an Apriori-based node-set enumeration in top-down process. However this assumption of prepared bR*-tree is not applicable to practical spatial web datasets, and the pruning ability of Apriori-based enumeration is highly dependent on the data distribution.　In this thesis, we do not expect any prepared data-partitioning, but assume that we create a grid partitioning from necessary data only when an mCK query is given. Under this assumption, we propose a new search strategy termed Diameter Candidate Check (DCC), which can find a smaller node-set at an earlier stage of search so that it can reduce search space more efficiently. According to DCC search strategy, we firstly employ an implementation of DCC strategy in a nested loop search algorithm (called DCC-NL). Next, we improve the DCC-NL in a recursive way (called RDCC). RDCC can afford a more reasonable priority order of node-set enumeration. We also uses a tight lower bound to improve pruning ability in RDCC.　RDCC performs well in a wide variey of data distributions, but it has still deficiency when one data-point has many query keywords and numerous node-sets are generated. Hence in order to avoid the generation of node-sets which is an unstable factor of search efficiency, we propose another different top-down search approach called Pairwise Expansion. Finally, we discuss some optimization techniques to enhance Pairwise Expansion approach. We first discuss the index structure in the Pairwise Expansion approach, and try to use an on-the-fly kd-tree to reduce building cost in the query process. Also a new lower bound and an upper bound are employed for more powerful pruning in Pairwise Expansion.　We evaluate these approaches by using both real datasets and synthetic datasets for different data distributions, including 1.6 million of Flickr photo data. The result shows that DCC strategy can provide more stable search performance than the Apriori-based approach. And the Pairwise Expansion approach enhanced with lower/upper bounds, has more advantages over those algorithms having node-set generation, and is applicable for real spatial web data.電気通信大学201

Probabilistic Analysis of Euclidean Capacitated Vehicle Routing

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(?n) or ?(?n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = ?n, the ITP algorithm is at best a (1+c?)-approximation for some positive constant c?. On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+? due to Bompadre, Dror, and Orlin, where ? is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+?. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest

The matching relaxation for a class of generalized set partitioning problems

This paper introduces a discrete relaxation for the class of combinatorial optimization problems which can be described by a set partitioning formulation under packing constraints. We present two combinatorial relaxations based on computing maximum weighted matchings in suitable graphs. Besides providing dual bounds, the relaxations are also used on a variable reduction technique and a matheuristic. We show how that general method can be tailored to sample applications, and also perform a successful computational evaluation with benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial Optimization), with proceedings on Electronic Notes in Discrete Mathematic

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions