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Register a new userComplexity of Cycle Length Modularity Problems in Graphs
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Mayur Thakur
Publication date
2003
fields
Informatics Engineering
and research's language is
English
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The even cycle problem for both undirected and directed graphs has been the topic of intense research in the last decade. In this paper, we study the computational complexity of emph{cycle length modularity problems}. Roughly speaking, in a cycle length modularity problem, given an input (undirected or directed) graph, one has to determine whether the graph has a cycle $C$ of a specific length (or one of several different lengths), modulo a fixed integer. We denote the two families (one for undirected graphs and one for directed graphs) of problems by $(S,m)hbox{-}{rm UC}$ and $(S,m)hbox{-}{rm DC}$, where $m in mathcal{N}$ and $S subseteq {0,1, ..., m-1}$. $(S,m)hbox{-}{rm UC}$ (respectively, $(S,m)hbox{-}{rm DC}$) is defined as follows: Given an undirected (respectively, directed) graph $G$, is there a cycle in $G$ whose length, modulo $m$, is a member of $S$? In this paper, we fully classify (i.e., as either polynomial-time solvable or as ${rm NP}$-complete) each problem $(S,m)hbox{-}{rm UC}$ such that $0 in S$ and each problem $(S,m)hbox{-}{rm DC}$ such that $0 otin S$. We also give a sufficient condition on $S$ and $m$ for the following problem to be polynomial-time computable: $(S,m)hbox{-}{rm UC}$ such that $0 otin S$.
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We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $Omega(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a $t$-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on $n$ which require $n+1$ queries classically and only $1$ query quantumly.
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