id,summary,reporter,owner,description,type,status,priority,milestone,component,version,resolution,keywords,cc,revision
605,Overflow in Optimal Cost,mgara,Alpar Juttner,"Motivation
==========
Please consider the following simple class of problems (which I will refer to
as the primal MRF problem [1]):
max_{x,r} c' r
wrt
r = |Ax - b|
where A is a network/graph matrix (the rows define the oriented edges, the
matrix is Totally Unimodular), and c >= 0, b are integral vectors. The dual of
this problem is an instance of an MCF problem defined on a graph represented
by the network matrix A with b as the costs of the edges and c as the
capacities. Strong duality holds here.
In our work, we are interested in examining and evaluating different approaches
to solving either the MRF or MCF problem using either primal, dual or
primal-dual methods. The context we have in mind is application to image
processing tasks, in particular we are interested in the phase unwrapping
problem in InSAR.
Properties of MCF Instances
===========================
1. The MCF dual problem of the MRF problem is defined on the same graph as
defined by A, with the addition of an opposite arc with negative cost for every
oriented arc in A.
2. The MCF dual problem has no node imbalances by definition. The network flow
condition must still be enforced.
3. By corollary of 1 & 2, the optimal values for the MCF dual problem must be
negative.
Current Issue with Lemon:
=========================
Please consider the attached code that was tested with Lemon version 1.3.1. It
should be self contained and compilable with the make command if the LEMON_HOME
directory is correctly set in the Makefile.
Once compiled it can be run as follows (assuming a linux environment with a
BASH shell) to reproduce the issue.
for scale in $(seq .1 .1 .4); do ./lemon_mcf_solver netgen_8_08a.txt $scale; done > output.txt
In output.txt we should see that as we scale the costs b by a factor of .1 - .4
smoothly we get a negative optimal value initially, but when we reach .4 we
obtain a positive optimal value:
grep ""simplex cost:\|scaling costs"" output.txt
INFO: scaling costs by 0.100000
INFO: network simplex cost: -575280232
INFO: scaling costs by 0.200000
INFO: network simplex cost: -1154018674
INFO: scaling costs by 0.300000
INFO: network simplex cost: -1732004546
INFO: scaling costs by 0.400000
INFO: network simplex cost: 1984124216
Given property 3 of these MCF instances we know that positive cost solution is
impossible to these MCF instances. In fact, we've verified that the solution is
correct up to scale .3 (by comparing with other solvers) and around the
scale of .4 is when we hit this issue.
We came across this issue by modifying some NETGEN instances to have the
properties of the MCF dual problem we expect.
Please let us know if this is an issue with how we are using the Lemon library,
or perhaps if this is a bug in the Lemon library that can be addressed.
Thank you in advance,
Matt
[1] Kolmogorov, Vladimir. ""Primal-dual algorithm for convex Markov random
fields."" Microsoft Research MSR-TR-2005-117 (2005).
",defect,closed,major,LEMON 1.4 release,core,hg main,invalid,,,