A time- and space-optimal algorithm for the many-visits TSP

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of $n$ cities that visits each city $c$ a prescribed number $k_c$ of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{\Theta(n)}$ space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time $2^{O(n)}$, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.Comment: Small fixes, journal versio

Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc $k$-Leaf-Out-Branching} problem we show that no polynomial kernel is possible unless polynomial hierarchy collapses to third level %$PH=\Sigma_p^3$ by applying a recent breakthrough result by Bodlaender et al. {BDFH08} in a non-trivial fashion. However our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable the {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding "cheat kernelization" raised in {IWPECOPEN08}

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

On width measures and topological problems on semi-complete digraphs

Under embargo until: 2021-02-01The topological theory for semi-complete digraphs, pioneered by Chudnovsky, Fradkin, Kim, Scott, and Seymour [10], [11], [12], [28], [43], [39], concentrates on the interplay between the most important width measures — cutwidth and pathwidth — and containment relations like topological/minor containment or immersion. We propose a new approach to this theory that is based on outdegree orderings and new families of obstacles for cutwidth and pathwidth. Using the new approach we are able to reprove the most important known results in a unified and simplified manner, as well as provide multiple improvements. Notably, we obtain a number of efficient approximation and fixed-parameter tractable algorithms for computing width measures of semi-complete digraphs, as well as fast fixed-parameter tractable algorithms for testing containment relations in the semi-complete setting. As a direct corollary of our work, we also derive explicit and essentially tight bounds on duality relations between width parameters and containment orderings in semi-complete digraphs.acceptedVersio

Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems

NP-hard problems have numerous applications in various fields such as networks, computer systems, circuit design, etc. However, no efficient algorithms have been found for NP-hard problems. It has been commonly believed that no efficient algorithms for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed that there are parameters much smaller than input sizes in many instances of NP-hard problems in the real world. In the last twenty years, researchers have been interested in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those instances with small parameters. Fixed-parameter tractable algorithms can practically find exact solutions to problem instances with small parameters, though those problems are considered intractable in traditional computational theory. In this dissertation, we propose a new approach of algorithm design and analysis: discovering better measures for problems. In particular we use two measures instead of the traditional single measure?input size to design algorithms and analyze their time complexity. For several classical NP-hard problems, we present improved algorithms designed and analyzed with this new approach, First we show that the new approach is extremely powerful for designing fixedparameter tractable algorithms by presenting improved fixed-parameter tractable algorithms for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected graph and the max-leaf problems on both directed and undirected graphs. Most of our algorithms are practical for problem instances with small parameters. Moreover, we show that this new approach is also good for designing exact algorithms (with no parameters) for NP-hard problems by presenting an improved exact algorithm for the well-known satisfiability problem. Our results demonstrate the power of this new approach to algorithm design and analysis for NP-hard problems. In the end, we discuss possible future directions on this new approach and other approaches to algorithm design and analysis

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum