Polynomial fixed-parameter algorithms : a case study for longest path on interval graphs.

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems

Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path; it is NP-hard in general but known to be solvable in O(n^4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k^9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis for polynomial-time solvable problems offers a very fertile ground for future studies for all sorts of algorithmic problems. It may enable a refined understanding of efficiency aspects for polynomial-time solvable problems, similarly to what classical parameterized complexity analysis does for NP-hard problems

The Maximum Binary Tree Problem

We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ? NP. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs

Faster Detours in Undirected Graphs

The $k$-Detour problem is a basic path-finding problem: given a graph $G$ on $n$ vertices, with specified nodes $s$ and $t$, and a positive integer $k$, the goal is to determine if $G$ has an $st$-path of length exactly $\text{dist}(s, t) + k$, where $\text{dist}(s, t)$ is the length of a shortest path from $s$ to $t$. The $k$-Detour problem is NP-hard when $k$ is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in $f(k)\text{poly}(n)$ time, for $f$ as slow-growing as possible. We present faster algorithms for $k$-Detour in undirected graphs, running in $1.853^k \text{poly}(n)$ randomized and $4.082^k \text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $2.746^k \text{poly}(n)$ randomized and $6.523^k \text{poly}(n)$ deterministic time [Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a "bipartitioned" subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length $k$ in undirected graphs [Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017]. Our work has direct implications for the $k$-Longest Detour problem: in this problem, we are given the same input as in $k$-Detour, but are now tasked with determining if $G$ has an $st$-path of length at least $\text{dist}(s, t) + k.$ Our results for k-Detour imply that we can solve $k$-Longest Detour in $3.432^k \text{poly}(n)$ randomized and $16.661^k \text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $7.539^k \text{poly}(n)$ randomized and $42.549^k \text{poly}(n)$ deterministic time [Fomin et al., STACS 2022]

More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width

Polynomial fixed-parameter algorithms : a case study for longest path on interval graphs.

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path, that is, given an undirected graph, find a maximum-length path in G. Longest Path is NP-hard in general but known to be solvable in O(n4) time on n-vertex interval graphs. We show how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k9n) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems

Multi-agent Path Planning and Network Flow

This paper connects multi-agent path planning on graphs (roadmaps) to network flow problems, showing that the former can be reduced to the latter, therefore enabling the application of combinatorial network flow algorithms, as well as general linear program techniques, to multi-agent path planning problems on graphs. Exploiting this connection, we show that when the goals are permutation invariant, the problem always has a feasible solution path set with a longest finish time of no more than $n + V - 1$ steps, in which $n$ is the number of agents and $V$ is the number of vertices of the underlying graph. We then give a complete algorithm that finds such a solution in $O(nVE)$ time, with $E$ being the number of edges of the graph. Taking a further step, we study time and distance optimality of the feasible solutions, show that they have a pairwise Pareto optimal structure, and again provide efficient algorithms for optimizing two of these practical objectives.Comment: Corrected an inaccuracy on time optimal solution for average arrival tim