### Run Time Analysis

The following is a brief sketch of the proof of the expected run time of this algorithm. For a formal proof please see:

Ken Clarkson's Las Vegas Algorithms for Linear and Integer programming when the dimension is small.

This sketch also assumes you've read the rest of this web page, it uses terms defined previously.
The run time analysis of these algorithms is based on two key lemmas:

**Lemma 1:** If V (defined in the algorithms) is nonempty it contains a constraint of S^{*}

**Lemma 2:** Let V^{*} be a subset of S, and let R be a random subset of S \ V^{*} of size r, with |S \ V^{*}| = n. Let V be the set of constraints violated by x^{*}(R union V^{*}). Then the expected size of V is no more than d(n - r + 1) / (r - d).

For proofs of these lemmas see Ken Clarkson's paper.

**Lemma 3:** The probability that |V| <= 2n^{1/2} is at least 1/2. So on average two executions are required to obtain a successful one.

This follows directly from lemma 2 and Markov's inequality, and substituting r by d(n^{1/2}). For the iterative case, take V^{*} = {}, and n = w(S), and it follows that it's expected to take on average two iterations to have w(V) <= 2w(S)/(9d - 1).

**Theorem 4** The iterative algorithm takes O(d^{2}n log n) + (d log n)O(d)^{d/2+O(1)}, expected time as n approaches infinity.

It can be shown that the loop body of the algorithm runs an expected O(d log n) times, by showing w(S^{*}) grows much faster than w(S), so after O(d log n) iterations either V is empty of w(S^{*}) > w(S) a contradiction. The Simplex Algorithm has a time bound of O(d)^{d/2 + O(1)}. Determining V takes time O(dn). So the body of the loop takes O(dn) + O(d)^{d/2+O(1)}. So bound on the theorem follows.

**Theorem 5** The mixed algorithm takes expected time of O(d^{2}n) + (d^{2}log n)O(d)^{d/2+O(1)} + O(d^{4}n^{1/2} log n) as n approaches infinity.

V^{*} grows by at most 2n^{1/2} each iteration where |V| <= 2n^{1/2}. So after at most d+1 such iterations, V^{*} will contain S^{*}. It follows that the maximum size of R union V^{*} is (9d^{2}n)^{1/2}. Let T_{m}(n) be the expected time of the mixed algorithm with n constraints and d variables. Similarily let T_{i}(n) be the expected time of the iterative algorithm. Since the probability that |V| <= 2n^{1/2} is at least 1/2, it follows that the expected run time of the mixed algorithm is:

T_{m}(n) <= 2(d+1)T_{i}((9d^{2}n)^{1/2}) + O(d^{2}n)

The last O(d^{2}n) term comes from checking the violatiors (O(dn) time) and expected 2(d + 1) times. The bound follows.