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Avi Wigderson studies computational complexity theory, including the role of randomized algorithms in efficient computation. His research connects pseudorandom generators, computational hardness and derandomization. His book Mathematics and Computation by Avi Wigderson explains how sufficiently strong circuit-hardness assumptions can supply generators that fool efficient computational tests.^[ref]

The mechanism turns a lower-bound assumption into an algorithmic resource. If a generator with a short seed preserves a randomized computation's acceptance probability closely enough, enumerating all seeds can replace sampling random bits. The seed must be short enough for that enumeration to meet the claimed time bound; simply fixing one arbitrary seed does not supply the guarantee. This makes the power of randomness a precise question about computation rather than a visual impression that an output “looks random”.

Directly scanning all $m$ edges to evaluate each child of each of $n$ choices takes $O(nm+n)$ elementary operations, in Big O notation. There is a faster implementation. When assigning vertex $v$, let $d_0$ and $d_1$ count its already assigned graph neighbours labelled zero and one. Contributions from edges not incident to $v$ are unchanged, and contributions from edges to unassigned graph neighbours stay $1/2$ for either choice. ThereforeChoose zero if $d_1\ge d_0$, otherwise one: put $v$ on the side with fewer already assigned graph neighbours. With adjacency lists, scanning each vertex's list once costs $O(n+m)$ operations overall and uses $O(n+m)$ storage including the graph. These bounds count operations on labels and counters that fit in machine words of $O(\log (n+m+2))$ bits. Multiplying the conditional expectation cut formula by two also permits exact integer comparisons without rounding.

If $\mathrm{OPT}$ is the size of a maximum cut, then $\mathrm{OPT}\le m$. The constructed cut of a graph thus has size at least $m/2\ge \mathrm{OPT}/2$, an approximation ratio of $1/2$ in the maximization convention. The five-versus-six example illustrates the gap between this guarantee and exact optimization.

For a general randomized algorithm, the averaging proof alone does not establish efficient derandomization. One needs a polynomial number of commitments, an efficient way to find a suitable child at each step, and manageable representations and evaluation costs for the relevant conditional expectations. Those evaluations can themselves be hard. Enumerating all completions verifies this small example, but would take exponentially many assignments as $n$ grows. The polynomial-time algorithm here comes from the local conditional expectation cut formula, not from that enumeration.

Order the edges arbitrarily and fix their colours one at a time. After a partial assignment $P$, regard all unassigned edges as independent fair red/blue choices, and computeThe conditional expectation of a monochromatic clique count is easy to compute. For each four-vertex subset $A$, let $r$ be the number of its six edges still unassigned. Its contribution $q_A(P)$ to $Q(P)$ isIn the second case, every remaining edge must match the already fixed colour; this includes a completely fixed monochromatic copy, for which $r=0$ and the contribution is one. In the last case, the six unassigned edges may become either all red or all blue. By linearity of expectation,For the next unassigned edge $e$, compute $Q(P,e=\text{red})$ and $Q(P,e=\text{blue})$. The law of total expectation givesChoose the colour with the smaller value, resolving a tie by a fixed rule. At least one is at most their average, so the loop invariant is that $Q$ never increases. Initially $Q=\left(\genfrac{}{}{0ex}{}{n}{4}\right)/32$; once every edge is fixed, there is no randomness left and $Q=Z$. Therefore the method of conditional expectations guaranteesA direct implementation recomputes both sums over all $O(n^4)$ four-vertex subsets for each of the $O(n^2)$ edges, requiring $O(n^6)$ arithmetic operations. The comparisons can be exact: every contribution is a multiple of $1/32$, so store $32Q$ as an integer with $O(\log n)$ bits. This gives $O(n^6\log n)$ bit complexity with elementary arithmetic, and hence a deterministic polynomial time construction. No random sampling is performed by this derandomization.

Valentine Kabanets is a computer scientist whose research concerns computational complexity theory, derandomization, pseudorandomness and circuit complexity. A recurring question in his research is when efficient randomized algorithms can be replaced by efficient deterministic algorithms.^[ref]

His research description emphasizes a connection between designing fast algorithms and proving circuit lower bounds. An upper bound supplies an efficient method for a problem; a lower bound rules out methods within a specified model. In some settings, progress on one side can imply progress on the other, turning apparent obstacles into tools.^[ref]

Identity source: www2.cs.sfu.ca/~kabanets/