We develop a new algebraic technique that gives a simple randomized algorithm for the simple $k$-path problem with the same complexity $O^*(1.657^k)$ as in [A. Bj\"orklund. Determinant Sums for Undirected Hamiltonicity. FOCS 2010, pp. 173--182, (2010). A. Bj\"orklund, T. Husfeldt, P. Kaski, M. Koivisto. Narrow sieves for parameterized paths and ... more >>>

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $\R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. ... more >>>

We study the success probability of the PPSZ algorithm on $(d,k)$-CSP formulas. We greatly simplify the analysis of Hertli, Hurbain, Millius, Moser, Szedlak, and myself for the notoriously difficult case that the input formula has more than one satisfying assignment.

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A Chor--Goldreich (CG) source [CG88] is a sequence of random variables $X = X_1 \circ \ldots \circ X_t$, each $X_i \sim \{0,1 \{^d$, such that each $X_i$ has $\delta d$ min-entropy for some constant $\delta > 0$, even conditioned on any fixing of $X_1 \circ \ldots \circ X_{i-1}$. We typically ... more >>>