In my Doxyfile (attached) I have:
OUTPUT_DIRECTORY =

The Doxyfile is in the root of the workspace and that is also where Doxygen is run.

If I leave "Folder from which doxygen is run" empty, I get the following stack trace:

FATAL: error
java.lang.NullPointerException
at hudson.plugins.doxygen.DoxygenDirectoryParser$2.invoke(DoxygenDirectoryParser.java:154)
at hudson.plugins.doxygen.DoxygenDirectoryParser$2.invoke(DoxygenDirectoryParser.java:152)
at hudson.FilePath.act(FilePath.java:788)
at hudson.FilePath.act(FilePath.java:770)
at hudson.plugins.doxygen.DoxygenDirectoryParser.getDoxygenGeneratedDir(DoxygenDirectoryParser.java:152)
at hudson.plugins.doxygen.DoxygenDirectoryParser.retrieveDoxygenDirectoryFromDoxyfile(DoxygenDirectoryParser.java:325)
at hudson.plugins.doxygen.DoxygenDirectoryParser.invoke(DoxygenDirectoryParser.java:54)
at hudson.plugins.doxygen.DoxygenDirectoryParser.invoke(DoxygenDirectoryParser.java:14)
at hudson.FilePath.act(FilePath.java:788)
at hudson.FilePath.act(FilePath.java:770)
at hudson.plugins.doxygen.DoxygenArchiver._perform(DoxygenArchiver.java:219)
at hudson.plugins.doxygen.DoxygenArchiver.perform(DoxygenArchiver.java:176)
at hudson.tasks.BuildStepMonitor$2.perform(BuildStepMonitor.java:27)
at hudson.model.AbstractBuild$AbstractRunner.perform(AbstractBuild.java:700)
at hudson.model.AbstractBuild$AbstractRunner.performAllBuildSteps(AbstractBuild.java:675)
at hudson.model.AbstractBuild$AbstractRunner.performAllBuildSteps(AbstractBuild.java:653)
at hudson.model.Build$RunnerImpl.post2(Build.java:162)
at hudson.model.AbstractBuild$AbstractRunner.post(AbstractBuild.java:622)
at hudson.model.Run.run(Run.java:1434)
at hudson.model.FreeStyleBuild.run(FreeStyleBuild.java:46)
at hudson.model.ResourceController.execute(ResourceController.java:88)
at hudson.model.Executor.run(Executor.java:238)
Build step 'Publish Doxygen' changed build result to FAILURE

If I set "Folder from which doxygen is run" to ".", then I get the following stacktrace:

Publishing Doxygen HTML results.
FATAL: error
hudson.AbortException: The output directory doesn't exist.
at hudson.plugins.doxygen.DoxygenDirectoryParser.retrieveDoxygenDirectoryFromDoxyfile(DoxygenDirectoryParser.java:343)
at hudson.plugins.doxygen.DoxygenDirectoryParser.invoke(DoxygenDirectoryParser.java:54)
at hudson.plugins.doxygen.DoxygenDirectoryParser.invoke(DoxygenDirectoryParser.java:14)
at hudson.FilePath.act(FilePath.java:788)
at hudson.FilePath.act(FilePath.java:770)
at hudson.plugins.doxygen.DoxygenArchiver._perform(DoxygenArchiver.java:219)
at hudson.plugins.doxygen.DoxygenArchiver.perform(DoxygenArchiver.java:176)
at hudson.tasks.BuildStepMonitor$2.perform(BuildStepMonitor.java:27)
at hudson.model.AbstractBuild$AbstractRunner.perform(AbstractBuild.java:703)
at hudson.model.AbstractBuild$AbstractRunner.performAllBuildSteps(AbstractBuild.java:678)
at hudson.model.AbstractBuild$AbstractRunner.performAllBuildSteps(AbstractBuild.java:656)
at hudson.model.Build$RunnerImpl.post2(Build.java:162)
at hudson.model.AbstractBuild$AbstractRunner.post(AbstractBuild.java:625)
at hudson.model.Run.run(Run.java:1435)
at hudson.model.FreeStyleBuild.run(FreeStyleBuild.java:46)
at hudson.model.ResourceController.execute(ResourceController.java:88)
at hudson.model.Executor.run(Executor.java:238)
Build step 'Publish Doxygen' changed build result to FAILURE