James Orlin | Massachusetts Institute of Technology (MIT) (original) (raw)
Papers by James Orlin
... Valentin Polishchuk Esther M. Arkin Kobus Barnard Kevin Coogan ... and Ramosof the Delaunay ... more ... Valentin Polishchuk Esther M. Arkin Kobus Barnard Kevin Coogan ... and Ramosof the Delaunay triangulation, convexity, and separability by a given line into sets of equal cardinality [1] (the tolerance of the Delaunay triangulation was independently studied by Weller [23]). ...
The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear progra... more The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, , !4''~ Ar estrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the :'*j choice of exiting variables. Using ideas from parametric linear programming, we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables.
Operations Research, Feb 9, 2009
Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search:... more Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search: The ACM Digital Library The Guide. ...
We consider the simplex algorithm as applied to minimum cost network flows on a directed graph G ... more We consider the simplex algorithm as applied to minimum cost network flows on a directed graph G = (V, E). First we consider the strongly convergent pivot rule of Elam, Glover, and Klingman as applied to generalized networks. We show that this pivot rule is equivalent to lexicography in its choice of the variable to leave the basis. We also show the following monotonicity property that is satisfied by each basis B of a generalized network flow problem. If b' b b* and if < B-lb', B-lb* < u, then 5 B-lb < u ; i.e., if a basis is feasible for b' and b* then it is feasible for b. Next we consider Dantzig's pivot rule of selecting the entering variable whose reduced cost is minimum and using lexicography to avoid cycling. We show that the maximum number of pivots using Dantzig's pivot.rule is O(IV12 EI logIVI) when applied to either the assignment problem or the shortest path problem. Moreover, the maximum number of consecutive degenerate pivots for the minimum cost network flow problem is O(VI JEJ loglVI).
In this paper we consider the minimum cost network flow problem:
A transshipment problem with demands that exceed network capacity can be solved by sending ow i n... more A transshipment problem with demands that exceed network capacity can be solved by sending ow i n s e v eral waves. How can this be done in the minimum number,T , o f w aves, and at minimum cost, if costs are piece-wise linear convex functions of the ow? In this paper, we show t h a t this problem can be solved using minfm log T log(m;U) 1+log(m;U);log(mU) g maximum ow computations and one minimum (convex) cost ow computation. Here m is the number of arcs, ; is the maximum supply or demand, and U is the maximum capacity. When there is only one sink, this problem can be solved in the same asymptotic time as one minimum (convex) cost ow computation. This improves upon the recent algorithm in 5] w h i c h solves the quickest transshipment problem (the above mentioned problem without costs) on k terminals using k log T maximum ow computations and k minimum cost ow computations. Our solutions start with a stationary fractional ow, as described in 5], and use rounding to transform this into an integral ow. The rounding procedure takes O(n) time.
We analyze the Unit Demand Euclidean Vehicle Routing Problem (VRP) where n customers are modeled ... more We analyze the Unit Demand Euclidean Vehicle Routing Problem (VRP) where n customers are modeled as uniformly i.i.d. points and have unit demand. We show new lower bounds on the optimal cost for the metric VRP and we analyze them in this setting. We prove that there exists a constantĉ > 0 such that the Iterated Tour Partitioning (ITP) heuristic given by Haimovich and Rinnooy Kan [9] is a 2 −ĉ approximation algorithm with probability arbitrarily close to one as the number of customers goes to infinity. It has been a long standing open problem whether one can improve upon the factor of 2 given in . We also generalize this result and previous ones to the multi-depot case.
In this paper, we present a new primal simplex pivot rule and analyse the worst-case complexity o... more In this paper, we present a new primal simplex pivot rule and analyse the worst-case complexity of the resulting simplex algorithm for the minium cost flow problem, the assignment problem and the shortest path problem. We consider networks with n nodes, m arcs, integral arc capacities bounded by an integer number U, and integral arc costs bounded by an integer number C. Let L and U denote the nonbasic arcs at their lower and upper bounds respectively, and cij denote the reduced cost of any arc (i, j). Further, let A be a parameter whose initial value is C. Then our pivot rule is as follows: Select as an entering arc any (i, j) L with cij < -A/2 or any (i, j) U with cij A/2; select the leaving arc so that the strong feasibility of the basis is maintained. When there is no nonbasic arc satisfying this rule then replace A by A/2. We show that the simplex algorithm using this rule performs O(nm U logC) pivots and can be implemented to run in O(m 2 U logC) time. Specializing these results for the assignment and shortest path problems we show that the simplex algorithm solves these problems in O(n 2 logC) pivots and O(nm logC) time. These algorithms use the same data structures that are typically used to implement the primal simplex algorithms for network problems and have enough flexibility for fine tuning the algorithms in practice. We also use these ideas to obtain an O(nm logC) label correcting algorithm for the shortest path problem with arbitrary arc lengths, and an improved implementation of Dantzig's pivot rule.
This paper consists of two parts, both of which address stability of perfect matchings. In the fi... more This paper consists of two parts, both of which address stability of perfect matchings. In the first part we consider instances of the Stable Roommates problem that arise from geometric representation of participants preferences: a participant is a point in Euclidean space, and his preference list is given by sorted distances to the other participants. We observe that, unlike in the general case, if there are no ties in the preference lists, there always exists a unique stable matching; a simple greedy algorithm finds the matching efficiently. We show that, also contrary to the general case, the problem admits polynomial-time solution even in the case when ties are present in the preference lists.
An integrated genomic physical map combines multiple sources of data to position landmarks and cl... more An integrated genomic physical map combines multiple sources of data to position landmarks and clones along a genome. "Path-based" strategies for constructing integrated maps employ paths of overlapping clones to bridge intervals between genetically mapped markers.
... Valentin Polishchuk Esther M. Arkin Kobus Barnard Kevin Coogan ... and Ramosof the Delaunay ... more ... Valentin Polishchuk Esther M. Arkin Kobus Barnard Kevin Coogan ... and Ramosof the Delaunay triangulation, convexity, and separability by a given line into sets of equal cardinality [1] (the tolerance of the Delaunay triangulation was independently studied by Weller [23]). ...
The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear progra... more The traditional perturbation (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore, , !4''~ Ar estrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the :'*j choice of exiting variables. Using ideas from parametric linear programming, we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables.
Operations Research, Feb 9, 2009
Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search:... more Google, Inc. (search), Subscribe (Full Service), Register (Limited Service, Free), Login. Search: The ACM Digital Library The Guide. ...
We consider the simplex algorithm as applied to minimum cost network flows on a directed graph G ... more We consider the simplex algorithm as applied to minimum cost network flows on a directed graph G = (V, E). First we consider the strongly convergent pivot rule of Elam, Glover, and Klingman as applied to generalized networks. We show that this pivot rule is equivalent to lexicography in its choice of the variable to leave the basis. We also show the following monotonicity property that is satisfied by each basis B of a generalized network flow problem. If b' b b* and if < B-lb', B-lb* < u, then 5 B-lb < u ; i.e., if a basis is feasible for b' and b* then it is feasible for b. Next we consider Dantzig's pivot rule of selecting the entering variable whose reduced cost is minimum and using lexicography to avoid cycling. We show that the maximum number of pivots using Dantzig's pivot.rule is O(IV12 EI logIVI) when applied to either the assignment problem or the shortest path problem. Moreover, the maximum number of consecutive degenerate pivots for the minimum cost network flow problem is O(VI JEJ loglVI).
In this paper we consider the minimum cost network flow problem:
A transshipment problem with demands that exceed network capacity can be solved by sending ow i n... more A transshipment problem with demands that exceed network capacity can be solved by sending ow i n s e v eral waves. How can this be done in the minimum number,T , o f w aves, and at minimum cost, if costs are piece-wise linear convex functions of the ow? In this paper, we show t h a t this problem can be solved using minfm log T log(m;U) 1+log(m;U);log(mU) g maximum ow computations and one minimum (convex) cost ow computation. Here m is the number of arcs, ; is the maximum supply or demand, and U is the maximum capacity. When there is only one sink, this problem can be solved in the same asymptotic time as one minimum (convex) cost ow computation. This improves upon the recent algorithm in 5] w h i c h solves the quickest transshipment problem (the above mentioned problem without costs) on k terminals using k log T maximum ow computations and k minimum cost ow computations. Our solutions start with a stationary fractional ow, as described in 5], and use rounding to transform this into an integral ow. The rounding procedure takes O(n) time.
We analyze the Unit Demand Euclidean Vehicle Routing Problem (VRP) where n customers are modeled ... more We analyze the Unit Demand Euclidean Vehicle Routing Problem (VRP) where n customers are modeled as uniformly i.i.d. points and have unit demand. We show new lower bounds on the optimal cost for the metric VRP and we analyze them in this setting. We prove that there exists a constantĉ > 0 such that the Iterated Tour Partitioning (ITP) heuristic given by Haimovich and Rinnooy Kan [9] is a 2 −ĉ approximation algorithm with probability arbitrarily close to one as the number of customers goes to infinity. It has been a long standing open problem whether one can improve upon the factor of 2 given in . We also generalize this result and previous ones to the multi-depot case.
In this paper, we present a new primal simplex pivot rule and analyse the worst-case complexity o... more In this paper, we present a new primal simplex pivot rule and analyse the worst-case complexity of the resulting simplex algorithm for the minium cost flow problem, the assignment problem and the shortest path problem. We consider networks with n nodes, m arcs, integral arc capacities bounded by an integer number U, and integral arc costs bounded by an integer number C. Let L and U denote the nonbasic arcs at their lower and upper bounds respectively, and cij denote the reduced cost of any arc (i, j). Further, let A be a parameter whose initial value is C. Then our pivot rule is as follows: Select as an entering arc any (i, j) L with cij < -A/2 or any (i, j) U with cij A/2; select the leaving arc so that the strong feasibility of the basis is maintained. When there is no nonbasic arc satisfying this rule then replace A by A/2. We show that the simplex algorithm using this rule performs O(nm U logC) pivots and can be implemented to run in O(m 2 U logC) time. Specializing these results for the assignment and shortest path problems we show that the simplex algorithm solves these problems in O(n 2 logC) pivots and O(nm logC) time. These algorithms use the same data structures that are typically used to implement the primal simplex algorithms for network problems and have enough flexibility for fine tuning the algorithms in practice. We also use these ideas to obtain an O(nm logC) label correcting algorithm for the shortest path problem with arbitrary arc lengths, and an improved implementation of Dantzig's pivot rule.
This paper consists of two parts, both of which address stability of perfect matchings. In the fi... more This paper consists of two parts, both of which address stability of perfect matchings. In the first part we consider instances of the Stable Roommates problem that arise from geometric representation of participants preferences: a participant is a point in Euclidean space, and his preference list is given by sorted distances to the other participants. We observe that, unlike in the general case, if there are no ties in the preference lists, there always exists a unique stable matching; a simple greedy algorithm finds the matching efficiently. We show that, also contrary to the general case, the problem admits polynomial-time solution even in the case when ties are present in the preference lists.
An integrated genomic physical map combines multiple sources of data to position landmarks and cl... more An integrated genomic physical map combines multiple sources of data to position landmarks and clones along a genome. "Path-based" strategies for constructing integrated maps employ paths of overlapping clones to bridge intervals between genetically mapped markers.