This question is based on some observations which could be wrong (In that case, let me know). I am applying to PhD program in various univ in US in the area of theoretical computer science. Some of the things that I hear is that getting admission in TCS in top 15 theory univ is tough. The toughness is obviously due to the large volumes of applications that these univ receive (dnt know how much?). However one of the big factors is limited funding available with prof. So who funds students? prof or univ? In some cases I heard, that it is the prof who funds the student, then in that case why the admission process is centralized (the prof who is actually funding may not be in the committee)?
I'm a theoretical computer scientist currently serving on the admissions committee of a large top-15 computer science department.
This year my department received about 50 PhD applications from people whose primary interest is theoretical computer science and probably another 50 with secondary interest in theory, out of 750 PhD applications overall. We've offered admission to about 10 theory PhD students (out of about 200 PhD offers total). We realistically expect three or four theory PhD students to accept our offer (out of about 80-100 total).
It's complicated. A typical PhD offer from a strong department includes guaranteed funding in some form. My department promises five years of funding to every incoming PhD student, assuming they make steady progress toward their degree. (Do not accept a PhD admission offer without funding. If they really want you, they'll pay for you.) Most of our students take 6 years to finish, but in practice, (100-ε)% of our students are funded for their entire stay. A typical theory student in my department is a TA for 2-4 semesters and an RA of fellow for the rest.
When a student is admitted, the department is making a contractual commitment to funding that student, assuming they make adequate progress toward their degree. In practice, most of that funding comes from individual faculty grants, most of the rest comes from the department's budget for teaching assistantships, and a small fraction comes from fellowships (university, NSF, DOE, NDSEG, Hertz, etc.).
The number of students that each group is allowed to admit depends primarily on three factors:
In US computer science programs, departments offer admission, not individual faculty. Students are completely free to change advisors or even research areas, even if their existing advisor is funding them. (Of course, they have to fulfill their funding obligations, but that's an orthogonal issue.) Formally, students in my department do not even choose their thesis advisors until the end of their first year. (One of my recent PhDs entered the department as an RA in distributed systems/sensor networks; he switched to algorithms at the end of his first year.) For that reason, it's crucial that the admissions decision does not rest entirely with a single faculty member.
There are several models for funding graduate students: often times the professor is responsible for funding. However, in many cases, the system has "joint" sponsorship—at first the students are sponsored by the department (while they do teaching service or are taking classes, for instance). After that, they are then paid for by their advisors.
The role of centralized admissions is to cut down on the cumulative workload. Especially with the ease of submitting applications electronically, if each professor were responsible for selecting his or her own students, faculty would be swamped by applications. Having a central pool makes the process simpler for everybody.
This is because the top universities (you said top 15) want to maintain their high standards of admission into the graduate program and would not generally want to allow a candidate with rather poor qualifications on paper into the program just because someone is willing to fund them.
Remember that a top university will also most likely have a very rigorous curriculum, which the student will have to successfully complete (at least in the US) before they can advance, and this is independent of the student's research work with the faculty that is willing to fund them. So if a candidate's qualifications do not convince the committee that they are capable of advancing the program after 2 years, they will most likely not admit them because it will then be a drain on the university's resources.
That said, it is possible for such candidates to still get in, but the faculty and their references will have to make a really strong case for them and they must have some redeeming quality/ability elsewhere.
Depending on the university, funding for students can be allocated in different ways. From what I've seen in Computer Science, you can be guaranteed funding, which usually means that your tuition will be covered to some extent and you may receive a living stipend. You can receive no funding, which means that you have to pay your own tuition and your own living expenses, or you can receive partial funding which is some subset of guaranteed funding.
Some universities or departments don't admit graduate students unless they are guaranteed funding by either the department or by a professor. In these cases, sometimes the department/school might have some money set aside to fund graduate students, usually as TAs, but professors will have their own funds through grants. This allows professors more latitude in choosing graduate students that they think are promising and who share the same research interests.
Other universities/departments will admit students without guaranteed funding. Students that are admitted without guaranteed funding will need to find their own funding sources through scholarships, fellowships, or finding their own paid positions (e.g., research assistants, project assistants, or teaching assistants). If the student is unable to find any of these, footing the bill for tuition will fall directly on them. This can be very stressful and can lead to students needing to drop out because they can't find funding or a mad scramble/funded positions being very competitive.
My observation, and your mileage may vary depending on university or department, is that if the university does not offer all graduate students guaranteed funding, they still try to limit the unfunded students that are admitted to be balanced against the number of funding opportunities that may become/be available. Departments also tend to admit slightly more students than they have positions for in anticipation of some students choosing to go to a different university after they receive all of their admittance letters.
As to why it's centralized, what people said about uniform standards and saving on administrative costs makes sense, and there's also an argument that many universities like to keep statistics and information on how many students are applying and being admitted to each department, what their demographics are, etc.