Robust optimization with incremental recourse

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

Improved Algorithms for Computing Fisher's Market Clearing Prices

ACM Digital Library url: http://dl.acm.org/citation.cfm?id=1806731We give the first strongly polynomial time algorithm for computing an equilibrium for the linear utilities case of Fisher's market model. We consider a problem with a set B of buyers and a set G of divisible goods. Each buyer i starts with an initial integral allocation ei of money. The integral utility for buyer i of good j is Uij. We first develop a weakly polynomial time algorithm that runs in O(n4 log Umax + n3 emax) time, where n = |B| + |G|. We further modify the algorithm so that it runs in O(n4 log n) time. These algorithms improve upon the previous best running time of O(n8 log Umax + n7 log emax), due to Devanur et al.United States. Office of Naval Research (ONR grant N00014-05-1-0165

A simple combinatorial algorithm for submodular function minimization

This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) [O (n superscript 6 E O log nM)] time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O ((n4EO+n5)log nM) [O ((n superscript 4 EO + n superscript 5) log nM)] time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) [O ((n superscript 5 EO + n superscript 6) log n)] time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.United States. Office of Naval Research ( ONR grant N00014-08-1-0029

Dynamic Programming Methodologies in Very Large Scale Neighborhood Search Applied to the Traveling Salesman Problem

We provide two different neighborhood construction techniques for creating exponentially large neighborhoods that are searchable in polynomial time using dynamic programming. We illustrate both of these approaches on very large scale neighborhood search techniques for the traveling salesman problem. Our approaches are intended both to unify previously known results as well as to offer schemas for generating additional exponential neighborhoods that are searchable in polynomial time. The first approach is to define the neighborhood recursively. In this approach, the dynamic programming recursion is a natural consequence of the recursion that defines the neighborhood. In particular, we show how to create the pyramidal tour neighborhood, the twisted sequences neighborhood, and dynasearch neighborhoods using this approach. In the second approach, we consider the standard dynamic program to solve the TSP. We then obtain exponentially large neighborhoods by selecting a polynomially bounded number of states, and restricting the dynamic program to those states only. We show how the Balas and Simonetti neighborhood and the insertion dynasearch neighborhood can be viewed in this manner. We also show that one of the dynasearch neighborhoods can be derived directly from the 2-exchange neighborhood using this approach

Analysis of a Proposed First Generation Physical Map of the Human Genome

Cohen and colleagues [1] recently described a project to characterize a human yeast artificial chromosome (YAC) library and offered a 'proposed data analysis strategy' that was said to yield a physical map covering 87% of the human genome. The authors provided no analytical evaluation to test the validity of their novel strategy for constructing 'paths' in the genome. We have now examined the proposed method in detail. Analytical studies show that most paths with at most two YACs or spanning less than 5 cM are valid, but most paths involving four or more YACs or spanning 5 cM or more are invalid. After restricting the map to paths with a high probability of being valid, we conclude that the remaining map properly covers at most 36% of the genome

Directed shortest paths via approximate cost balancing

We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an O(m √ n log n) preprocessing step. Using these reduced costs, every shortest path query can be solved in O(m) time using an adaptation of Thorup’s component hierarchy method. The balancing result can also be applied to the ℓ∞-matrix balancing problem

A Computationally Efficient FPTAS for Convex Stochastic Dynamic Programs

We propose a computationally efficient fully polynomial-time approximation scheme (FPTAS) to compute an approximation with arbitrary precision of the value function of convex stochastic dynamic programs, using the technique of K-approximation sets and functions introduced by Halman et al. [Math. Oper. Res., 34, (2009), pp. 674-685]. This paper deals with the convex case only, and it has the following contributions. First, we improve on the worst-case running time given by Halman et al. Second, we design and implement an FPTAS with excellent computational performance and show that it is faster than an exact algorithm even for small problem instances and small approximation factors, becoming orders of magnitude faster as the problem size increases. Third, we show that with careful algorithm design, the errors introduced by floating point computations can be bounded, so that we can provide a guarantee on the approximation factor over an exact infinite-precision solution. We provide an extensive computational evaluation based on randomly generated problem instances coming from applications in supply chain management and finance. The running time of the FPTAS is both theoretically and experimentally linear in the size of the uncertainty set

Sensitivity Analysis for Shortest Path Problems and Maximum Capacity Path Problems in Undirected Graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path

The Infinite-Horizon Dynamic Lot-Size Problem with Cyclic Demand and Costs

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