arXiv subjects

Ei Ando

Publications and source records attributed to Ei Ando.

3 recordsLinked to original sources

arXiv2026

The Complexity of Computing Path Length Distributions with Edges i.i.d. Random via Local Uniformity

We investigate the problem of computing the distribution function for the shortest and longest path lengths in a directed graph with random edge lengths. Specifically, when these lengths are uniformly distributed, the problem reduces to computing the volume of a polytope defined by the graph structure. We establish that the problem is $\#P$-hard, even under the restricted condition that the random edge lengths are identically and independently distributed (i.i.d.) according to any continuous probability distribution with certain natural conditions, the local uniformity. This hardness result applies broadly: while the uniform distribution provides an essential case for the reduction, other distributions -- such as exponential or normal -- are similarly hard because they contain uniform distributions in every arbitrarily small interval. Furthermore, we show that the problem is contained within $\mathrm{XP}$ with respect to the treewidth $k$ of the underlying undirected graph. For the specific case of i.i.d. uniform edge lengths, we present a novel dynamic programming algorithm that processes a tree decomposition by iteratively performing convolutions to propagate distribution functions. Our approach achieves a time complexity of $n^{O(k^2)}$ for any fixed treewidth $k$.

cs.CC↗

arXiv2022

The Distribution Function of the Longest Path Length in Constant Treewidth DAGs with Random Edge Length

This paper is about the length $X_{\rm MAX}$ of the longest path in directed acyclic graph (DAG) $G=(V,E)$ with random edge lengths, where $|V|=n$ and $|E|=m$. When the edge lengths are mutually independent and uniformly distributed, the problem of computing the distribution function $\Pr[X_{\rm MAX}\le x]$ is known to be $\#$P-hard even in case $G$ is a directed path. In this case, $\Pr[X_{\rm MAX}\le x]$ is equal to the volume of the knapsack polytope, an $m$-dimensional unit hypercube truncated by a halfspace. In this paper, we show that there is a deterministic fully polynomial time approximation scheme (FPTAS) for computing $\Pr[X_{\rm MAX}\le x]$ in case the treewidth of $G$ is at most a constant $k$. The running time of our algorithm is $O(k^2 n(\frac{16(k+1)mn^2}ε)^{4k^2+6k+2})$ to achieve a multiplicative approximation ratio $1+ε$. Before our FPTAS, we present a fundamental formula that represents $\Pr[X_{\rm MAX}\le x]$ by at most $n-1$ repetitions of definite integrals. Moreover, in case the edge lengths follow the mutually independent standard exponential distribution, we show a $((4k+2)mn)^{O(k)}$ time exact algorithm. For random edge lengths satisfying certain conditions, we also show that computing $\Pr[X_{\rm MAX}\le x]$ is fixed parameter tractable if we choose treewidth $k$, the additive error $ε'$, and $x$ as the parameters.

cs.CC↗

arXiv2016

An FPTAS for the Volume of a ${\cal V}$-polytope ---It is Hard to Compute The Volume of The Intersection of Two Cross-polytopes

Given an $n$-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio $(n/\log n)^n$. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a deterministic approximation of the volume of a ${\cal V}$-polytope, that is a polytope with few vertices and (possibly) exponentially many facets, this paper investigates the volume of a "knapsack dual polytope," which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, the volume of the intersection of two cross-polytopes (i.e., $L_1$-balls) is #P-hard, unlike the cases of $L_{\infty}$-balls or $L_2$-balls.

cs.CC↗