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

Faster Deterministic Algorithms for Packing, Matching and tt-Dominating Set Problems

In this paper, we devise three deterministic algorithms for solving the $m$-set $k$-packing, $m$-dimensional $k$-matching, and $t$-dominating set problems in time $O^*(5.44^{mk})$, $O^*(5.44^{(m-1)k})$ and $O^*(5.44^{t})$, respectively. Although recently there has been remarkable progress on randomized solutions to those problems, our bounds make good improvements on the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with arXiv:1303.047

Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by $p$. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by $p$. We complete the picture here by our main result which is that both \textsc{Directed Vertex Multiway Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time $2^{2^{O(p)}}n^{O(1)}$, i.e., FPT parameterized by size $p$ of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that \textsc{Directed Multicut} is FPT for the case of $k=2$ terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011)

Chain minors are FPT

Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain C_Q in Q with the property that f restricted to C_Q is an isomorphism of chains. We give an algorithm to decide whether a poset P is a chain minor of o poset Q that runs in time O(|Q| log |Q|) for every fixed poset P. This solves an open problem from the monograph by Downey and Fellows [Parameterized Complexity, 1999] who asked whether the problem was fixed parameter tractable

Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires n^{epsilon k} time, for some constant epsilon>1/2, to count (note that these conjectures are significantly weaker than the usual ones made on these problems) on randomized machines for all but finitely many input lengths, then we have the following derandomizations: - BPP can be decided in polynomial time using only n^alpha random bits on average over any efficient input distribution, for any constant alpha>0 - BPP can be decided in polynomial time with no randomness on average over the uniform distribution This answers an open question of Ball et al. (STOC \u2717) in the positive of whether derandomization can be achieved from conjectures from fine-grained complexity theory. More strongly, these derandomizations improve over all previous ones achieved from worst-case uniform assumptions by succeeding on all but finitely many input lengths. Previously, derandomizations from worst-case uniform assumptions were only know to succeed on infinitely many input lengths. It is specifically the structure and moderate hardness of the k-Orthogonal Vectors and k-CLIQUE problems that makes removing this restriction possible. Via this uniform derandomization, we connect the problem-centric and resource-centric views of complexity theory by showing that exact hardness assumptions about specific problems like k-CLIQUE imply quantitative and qualitative relationships between randomized and deterministic time. This can be either viewed as a barrier to proving some of the main conjectures of fine-grained complexity theory lest we achieve a major breakthrough in unconditional derandomization or, optimistically, as route to attain such derandomizations by working on very concrete and weak conjectures about specific problems