17,475 research outputs found
Two new algorithms for the linear assignment problem
Ankara : Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent University, 1990.Thesis (Master's) -- Bilkent University, 1990.Includes bibliographical references.The linear assignment problem (AP) being among the first linear programming problems
to be studied extensively,, is a fundamental problem in combinatorial optimization and
network flow theory. AP arises in numerous applications of assigning personnel to jobs,
assigning facilities to locations, sequencing jobs, scheduling flights, project planning and
a variety of other practica.1 problems in logistics planning. In this thesis work, we seek for
new approaches for solving the linear assignment problem. The main concern is to develop
solution methods that exhibit some sort of parallelism. We present two new approaches
for solving the assignment problem : A dual-feasible signature guided forest algorithm and
a criss-cross like algorithm.Ekin, OyaM.S
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
Some NP-complete edge packing and partitioning problems in planar graphs
Graph packing and partitioning problems have been studied in many contexts,
including from the algorithmic complexity perspective. Consider the packing
problem of determining whether a graph contains a spanning tree and a cycle
that do not share edges. Bern\'ath and Kir\'aly proved that this decision
problem is NP-complete and asked if the same result holds when restricting to
planar graphs. Similarly, they showed that the packing problem with a spanning
tree and a path between two distinguished vertices is NP-complete. They also
established the NP-completeness of the partitioning problem of determining
whether the edge set of a graph can be partitioned into a spanning tree and a
(not-necessarily spanning) tree. We prove that all three problems remain
NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure
Submodular linear programs on forests
A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed
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