1,600 research outputs found
The Complexity of the Simplex Method
The simplex method is a well-studied and widely-used pivoting method for
solving linear programs. When Dantzig originally formulated the simplex method,
he gave a natural pivot rule that pivots into the basis a variable with the
most violated reduced cost. In their seminal work, Klee and Minty showed that
this pivot rule takes exponential time in the worst case. We prove two main
results on the simplex method. Firstly, we show that it is PSPACE-complete to
find the solution that is computed by the simplex method using Dantzig's pivot
rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a
specific variable to enter the basis is PSPACE-complete. We use the known
connection between Markov decision processes (MDPs) and linear programming, and
an equivalence between Dantzig's pivot rule and a natural variant of policy
iteration for average-reward MDPs. We construct MDPs and show
PSPACE-completeness results for single-switch policy iteration, which in turn
imply our main results for the simplex method
Edge Elimination in TSP Instances
The Traveling Salesman Problem is one of the best studied NP-hard problems in
combinatorial optimization. Powerful methods have been developed over the last
60 years to find optimum solutions to large TSP instances. The largest TSP
instance so far that has been solved optimally has 85,900 vertices. Its
solution required more than 136 years of total CPU time using the
branch-and-cut based Concorde TSP code [1]. In this paper we present graph
theoretic results that allow to prove that some edges of a TSP instance cannot
occur in any optimum TSP tour. Based on these results we propose a
combinatorial algorithm to identify such edges. The runtime of the main part of
our algorithm is for an n-vertex TSP instance. By combining our
approach with the Concorde TSP solver we are able to solve a large TSPLIB
instance more than 11 times faster than Concorde alone
Avoiding unnecessary demerging and remerging of multiâcommodity integer flows
Resource flows may merge and demerge at a network node. Sometimes several demerged flows may be immediately merged again, but in different combinations compared to before they were demerged. However, the demerging is unnecessary in the first place if the total resources at each of the network nodes involved remains unchanged. We describe this situation as âunnecessary demerging and remerging (UDR)â of flows, which would incur unnecessary operations and costs in practice. Multiâcommodity integer flows in particular will be considered in this paper. This deficiency could be theoretically overcome by means of fixedâcharge variables, but the practicality of this approach is restricted by the difficulty in solving the corresponding integer linear program (ILP). Moreover, in a problem where the objective function has many cost elements, it would be helpful if such operational costs are optimized implicitly. This paper presents a heuristic branching method within an ILP solver for removing UDR without the use of fixedâcharge variables. We use the concept of âflow potentialsâ (different from âflow residuesâ for maxâflows) guided by which underutilized arcs are heuristically banned, thus reducing occurrences of UDR. Flow connection bigraphs and flow connection groups (FCGs) are introduced. We prove that if certain conditions are met, fully utilizing an arc will guarantee an improvement within an FCG. Moreover, a location subâmodel is given when the former cannot guarantee an improvement. More importantly, the heuristic approach can significantly enhance the full fixedâcharge model by warmâstarting. Computational experiments based on realâworld instances have shown the usefulness of the proposed methods
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
On the Solution of Linear Programming Problems in the Age of Big Data
The Big Data phenomenon has spawned large-scale linear programming problems.
In many cases, these problems are non-stationary. In this paper, we describe a
new scalable algorithm called NSLP for solving high-dimensional, non-stationary
linear programming problems on modern cluster computing systems. The algorithm
consists of two phases: Quest and Targeting. The Quest phase calculates a
solution of the system of inequalities defining the constraint system of the
linear programming problem under the condition of dynamic changes in input
data. To this end, the apparatus of Fejer mappings is used. The Targeting phase
forms a special system of points having the shape of an n-dimensional
axisymmetric cross. The cross moves in the n-dimensional space in such a way
that the solution of the linear programming problem is located all the time in
an "-vicinity of the central point of the cross.Comment: Parallel Computational Technologies - 11th International Conference,
PCT 2017, Kazan, Russia, April 3-7, 2017, Proceedings (to be published in
Communications in Computer and Information Science, vol. 753
On the Complexity of an Unregulated Traffic Crossing
The steady development of motor vehicle technology will enable cars of the
near future to assume an ever increasing role in the decision making and
control of the vehicle itself. In the foreseeable future, cars will have the
ability to communicate with one another in order to better coordinate their
motion. This motivates a number of interesting algorithmic problems. One of the
most challenging aspects of traffic coordination involves traffic
intersections. In this paper we consider two formulations of a simple and
fundamental geometric optimization problem involving coordinating the motion of
vehicles through an intersection.
We are given a set of vehicles in the plane, each modeled as a unit
length line segment that moves monotonically, either horizontally or
vertically, subject to a maximum speed limit. Each vehicle is described by a
start and goal position and a start time and deadline. The question is whether,
subject to the speed limit, there exists a collision-free motion plan so that
each vehicle travels from its start position to its goal position prior to its
deadline.
We present three results. We begin by showing that this problem is
NP-complete with a reduction from 3-SAT. Second, we consider a constrained
version in which cars traveling horizontally can alter their speeds while cars
traveling vertically cannot. We present a simple algorithm that solves this
problem in time. Finally, we provide a solution to the discrete
version of the problem and prove its asymptotic optimality in terms of the
maximum delay of a vehicle
Nonclassicality of pure two-qutrit entangled states
We report an exhaustive numerical analysis of violations of local realism by
two qutrits in all possible pure entangled states. In Bell type experiments we
allow any pairs of local unitary U(3) transformations to define the measurement
bases. Surprisingly, Schmidt rank-2 states, resembling pairs of maximally
entangled qubits, lead to the most noise-robust violations of local realism.
The phenomenon seems to be even more pronounced for four and five dimensional
systems, for which we tested a few interesting examples.Comment: 6 pages, journal versio
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms
We give a lower bound on the iteration complexity of a natural class of
Lagrangean-relaxation algorithms for approximately solving packing/covering
linear programs. We show that, given an input with random 0/1-constraints
on variables, with high probability, any such algorithm requires
iterations to compute a
-approximate solution, where is the width of the input.
The bound is tight for a range of the parameters .
The algorithms in the class include Dantzig-Wolfe decomposition, Benders'
decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for
lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988]
and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy
argument to show an analogous lower bound on the support size of
-approximate mixed strategies for random two-player zero-sum
0/1-matrix games
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