I have access to a company\'s unique linked in ID and would like to be able to do a search for its job postings; however, the only company-related input for the job-search
Linkedin does not expose most of its apis publicly anymore starting from May, 2015. Please go through this link (https://developer.linkedin.com/blog/posts/2015/developer-program-changes) for more info. You cannot achieve this anymore unless you are a part of their partnership program. More info here https://developer.linkedin.com/partner-programs
You can do this a couple of different ways. In neither case do you need to
Using job-search as you noted, you can do this:
http://api.linkedin.com/v1/job-search:(jobs:(id,company:(id,name),position:(title,location,job-functions))&company-name=LinkedIn
Note that job-search is a fuzzy match in this case so if you're close, it's probably good enough. You don't want to include "Inc" if you're looking for a company, though. The documentation for the job-search API is here: https://developer.linkedin.com/documents/job-search-api
Check the Hacker Summary at the top of the page to see all the different fields you can retrieve with a single call (there's no need to make subsequent calls to get more details).
However, if you've gotten the company ID you can get jobs directly from the job lookup API: https://developer.linkedin.com/documents/job-lookup-api-and-fields
http://api.linkedin.com/v1/jobs/1337:(id,company:(id,name),position:(title,location,job-functions)
Again, the same field selectors apply here. You can request the details for the job in the same query you use to find the jobs in the first place.
LinkedIn API for jobs of particular company based on company Id http://api.linkedin.com/v1/companies/(LinkediUniqueCompanyID)/updates?event-type=job-posting
LinkedIn may someday improve their API to handle requests such as the one needed here. But for now, it looks like the only way to get all job postings for a company based on the company ID is to make 2 separate preliminary calls (one to acquire the company name using the company ID, then a second to acquire all jobs based on the company name). Then another call to the API must be made for each job ID returned by the second call.
I know this isn't the API but you might find this plugin that they provide to be useful:
https://developer.linkedin.com/plugins/jobs-you-may-be-interested?button-type-yours=true
Just put in your company name and it gives a nice widget with your company's jobs. Again, not really the API but I thought it might accomplish what you wanted anyways.