Relations between automata and the simple k-path problem

Let $G$ be a directed graph on $n$ vertices. Given an integer $k<=n$, the SIMPLE $k$-PATH problem asks whether there exists a simple $k$-path in $G$. In case $G$ is weighted, the MIN-WT SIMPLE $k$-PATH problem asks for a simple $k$-path in $G$ of minimal weight. The fastest currently known deterministic algorithm for MIN-WT SIMPLE $k$-PATH by Fomin, Lokshtanov and Saurabh runs in time $O(2.851^k\cdot n^{O(1)}\cdot \log W)$ for graphs with integer weights in the range $[-W,W]$. This is also the best currently known deterministic algorithm for SIMPLE k-PATH- where the running time is the same without the $\log W$ factor. We define $L_k(n)\subseteq [n]^k$ to be the set of words of length $k$ whose symbols are all distinct. We show that an explicit construction of a non-deterministic automaton (NFA) of size $f(k)\cdot n^{O(1)}$ for $L_k(n)$ implies an algorithm of running time $O(f(k)\cdot n^{O(1)}\cdot \log W)$ for MIN-WT SIMPLE $k$-PATH when the weights are non-negative or the constructed NFA is acyclic as a directed graph. We show that the algorithm of Kneis et al. and its derandomization by Chen et al. for SIMPLE $k$-PATH can be used to construct an acylic NFA for $L_k(n)$ of size $O^*(4^{k+o(k)})$. We show, on the other hand, that any NFA for $L_k(n)$ must be size at least $2^k$. We thus propose closing this gap and determining the smallest NFA for $L_k(n)$ as an interesting open problem that might lead to faster algorithms for MIN-WT SIMPLE $k$-PATH. We use a relation between SIMPLE $k$-PATH and non-deterministic xor automata (NXA) to give another direction for a deterministic algorithm with running time $O^*(2^k)$ for SIMPLE $k$-PATH

Finding the Minimum-Weight k-Path

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M). If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of \tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k) M n^3) and a $(1+\varepsilon)$-approximate solution of running time \tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201

Monomial Testing and Applications

In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The other is an $O^*(12.8^k)$ time deterministic algorithm for the same $q$-monomial testing problem but requiring the polynomials to be represented by tree-like circuits. Several applications of $q$-monomial testing are also given, including a deterministic $O^*(12.8^{mk})$ upper bound for the $m$-set $k$-packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note: substantial text overlap with arXiv:1302.5898; and text overlap with arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor