k-Path Problem Research Articles

Deciding path size of nondeterministic (and input-driven) pushdown automata

The degree of ambiguity (respectively, the path size) of a nondeterministic automaton, on a given input, measures the number of accepting computations (respectively, the number of all computations). It is known that deciding the finiteness of the degree of ambiguity of a nondeterministic pushdown automaton is undecidable. Also, it is undecidable for a given k≥3 to decide whether the path size of a nondeterministic pushdown automaton is bounded by k.As the main result, we show that deciding the finiteness of the path size of a nondeterministic pushdown automaton can be done in polynomial time. Also, we show that the k-path problem for nondeterministic input-driven pushdown automata (respectively, for nondeterministic finite automata) is complete for exponential time (respectively, complete for polynomial space).

A note on algebraic techniques for subgraph detection

The k-path problem asks whether a given graph contains a simple path of length k. Along with other prominent parameterized problems, it reduces to the problem of detecting multilinear terms of degree k (k-MLD), making the latter a fundamental problem in parameterized algorithms.This has generated significant efforts directed at devising fast deterministic algorithms for k-MLD, and there are now at least two independent approaches that yield the same record bound on the running time. Namely the combinatorial representative-set based approach of Fomin et al. (JACM'16), and the algebraic techniques of Pratt (FOCS'19) and Brand and Pratt (ICALP'21).In this note, we explore the relationship between the latter results, based on partial differentials, and a previous algebraic approach based on the exterior algebra (Brand, ESA'19; Brand, Dell and Husfeldt, STOC'18). We do so by studying the relevant algebraic objects. These are on the one hand (1) the subalgebras of the tensor square of the exterior algebra generated in degree one. On the other hand, we consider (2) the space of partial derivatives of generic determinants, and closely related, (3) the space of minors of generic matrices.We prove that (2) arises as a quotient of (1), and that there is an isomorphism between the objects (1) and (3). Hence, the techniques are essentially equivalent, and the quotient relation between (2) and both of (1) and (3) hints at a possible refinement of the techniques.

Faster deterministic parameterized algorithm for k-Path

In the k-Path problem, the input is a directed graph G and an integer k≥1, and the goal is to decide whether there is a simple directed path in G with exactly k vertices. We give a deterministic algorithm for k-Path with time complexity O⁎(2.554k). This improves the previously best deterministic algorithm for this problem of Zehavi [ESA 2015] whose time complexity is O⁎(2.597k). The technique used by our algorithm can also be used to obtain faster deterministic algorithms for k-Tree, r-Dimensionalk-Matching, Graph Motif, and Partial Cover.

On the kernelization of split graph problems

A split graph is a graph whose vertices can be partitioned into a clique and an independent set. We study numerous problems on split graphs, namely the k-Vertex-Disjoint Paths, k-Cycle, k-Path and k-ℓ-Stable Set problems. In the k-Vertex-Disjoint Paths problem, we are given a graph and k terminal pairs of vertices, and are asked whether there is a set of k vertex-disjoint paths linking these terminal pairs, respectively. In the k-Cycle/k-Path problem, we are given a graph and are asked whether there is a path/cycle of length k. The k-ℓ-Stable Set problem takes a graph and an integer k as input, and asks whether the graph has a subset of k vertices such that the distance between every two vertices in the subset is at least ℓ+1. It is known that all the above problems are NP-complete on split graphs. We derive a 4k-vertex kernel for the k-Vertex-Disjoint Paths problem and an O(k2)-vertex kernel for both the k-Path problem and the k-Cycle problem. Concerning the k-ℓ-Stable Set problem, for ℓ=1 or ℓ≥3, the problem is polynomial-time solvable on split graphs. For ℓ=2, we prove that the k-ℓ-Stable Set problem is W[1]-complete on split graphs, with respect to k. However, if the given split graph contains no K1,r as an induced subgraph, and every vertex in the independent set of the split graph has degree at most d, we derive a linear vertex kernel for the k-2-Stable Set problem, where both r and d are constants.

Narrow sieves for parameterized paths and packings

We present parameterized algorithms for the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space. The constant bases of the exponentials are significantly smaller than in previous works; for example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with d colors in time within a polynomial factor of 2(d−1)n/2. Our techniques generalize an algebraic approach studied in various recent works.

Turing kernelization for finding long paths and cycles in restricted graph classes

The k-Path problem asks whether a given undirected graph has a (simple) path of length k. We prove that k-Path has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to K3,t-minor-free graphs. This means that there is an algorithm that, given a k-Path instance (G,k) belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves k-Path instances of size polynomial in k in a single step. Our techniques also apply to k-Cycle, which asks for a cycle of length at least k.

A randomized algorithm for long directed cycle

Given a directed graph G and a parameter k, the Long Directed Cycle (LDC) problem asks whether G contains a simple cycle on at least k vertices, while the k-Path problem asks whether G contains a simple path on exactly k vertices. Given a deterministic (randomized) algorithm for k-Path as a black box, which runs in time t(G,k), we prove that LDC can be solved in deterministic time O⁎(max⁡{t(G,2k),4k+o(k)}) or in randomized time O⁎(max⁡{t(G,2k),4k}). In particular, we get that LDC can be solved in randomized time O⁎(4k).

REMARK ON GENERALIZED UNIVERSAL COVERING SPACE IN DIGITAL COVERING THEORY

As a survey-type article, the paper reviews the recent results on a (generalized) universal covering space in digital covering theory. The recent paper [19] established the generalized universal (2, k)-covering property which improves the universal (2, k)-covering property of [3]. In algebraic topology it is well-known that a simply connected and locally path connected covering space is a universal covering space. Unlike this property, in digital covering theory we can propose that a generalized universal covering space has its intrinsic feature. This property can be useful in classifying digital covering spaces and in studying a shortest k-path problem in data structure.