arXiv subjects

Jannis Blauth

Publications and source records attributed to Jannis Blauth.

8 recordsLinked to original sources

arXiv2025

Toward Optimal Approximations for Resource-Minimization for Fire Containment on Trees and Non-Uniform k-Center

One of the most elementary spreading models on graphs can be described by a fire spreading from a burning vertex in discrete time steps. At each step, all neighbors of burning vertices catch fire. A well-studied extension to model fire containment is to allow for fireproofing a number $B$ of non-burning vertices at each step. Interestingly, basic computational questions about this model are computationally hard even on trees. One of the most prominent such examples is Resource Minimization for Fire Containment (RMFC), which asks how small $B$ can be chosen so that a given subset of vertices will never catch fire. Despite recent progress on RMFC on trees, prior work left a significant gap in terms of its approximability. We close this gap by providing an optimal $2$-approximation and an asymptotic PTAS, resolving two open questions in the literature. Both results are obtained in a unified way, by first designing a PTAS for a smooth variant of RMFC, which is obtained through a careful LP-guided enumeration procedure. Moreover, we show that our new techniques, with several additional ingredients, carry over to the non-uniform $k$-center problem (NUkC), by exploiting a link between RMFC on trees and NUkC established by Chakrabarty, Goyal, and Krishnaswamy. This leads to the first approximation algorithm for NUkC that is optimal in terms of the number of additional centers that have to be opened.

cs.DS↗

arXiv2025

A Constant-Factor Approximation for Directed Latency

In the Directed Latency problem, we are given an asymmetric metric space $(V \cup \{s\},c)$ on a set $V$ of clients and a depot $s$. We are looking for a path $P$ starting in $s$ that visits all clients and minimizes the sum of the clients' waiting times (also known as latency) before being visited on the path. In contrast to the symmetric version of this problem (also known as the Deliveryperson problem and the Repairperson problem in the literature), there are significant gaps in our understanding of Directed Latency. The best approximation factor has remained at $O(\log |V|)$, as shown by [Friggstad, Salavatipour, and Svitkina, '13], for more than a decade. Only recently, [Friggstad and Swamy, '22] presented a constant-factor approximation but in quasi-polynomial time. Both results follow similar ideas: they consider buckets with geometrically increasing distances, build a path on each bucket, and then stitch together all these paths to get a feasible solution. [Friggstad and Swamy, '22] showed that by guessing a vertex from each bucket and augmenting a standard LP relaxation with these guesses, one can reduce the stitching cost. Unfortunately, the number of buckets is logarithmic in the number of vertices, so the running time of their algorithm is quasi-polynomial. In this paper, we present the first constant-factor approximation for Directed Latency in polynomial time by introducing a completely new way of bucketing, which helps us strengthen a standard LP relaxation with less aggressive guessing. Although the resulting LP is no longer a relaxation of Directed Latency, it still admits a good solution. We present a rounding algorithm for fractional solutions of our LP, crucially exploiting the way we restricted the feasibility region of the LP formulation.

cs.DS↗

arXiv2024

Cost Allocation for Set Covering: the Happy Nucleolus

We consider cost allocation for set covering problems. We allocate as much cost to the elements (players) as possible without violating the group rationality condition (no subset of players pays more than covering this subset would cost), and so that the excess vector is lexicographically maximized. This is identical to the well-known nucleolus if the core of the corresponding cooperative game is nonempty, i.e., if some optimum fractional cover is integral. In general, we call this the 'happy nucleolus'. Like for the nucleolus, the excess vector contains an entry for every subset of players, not only for the sets in the given set covering instance. Moreover, it is NP-hard to compute a single entry because this requires solving a set covering problem. Nevertheless, we give an explicit family of at most $mn$ subsets, each with a trivial cover (by a single set), such that the happy nucleolus is always completely determined by this proxy excess vector; here $m$ and $n$ denote the number of sets and the number of players in our set covering instance. We show that this is the unique minimal such family in a natural sense. While computing the nucleolus for set covering is NP-hard, our results imply that the happy nucleolus can be computed in polynomial time.

math.CO↗

arXiv2023

Improved guarantees for the a priori TSP

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space $(V,c)$ and an activation probability $p(v)$ for each customer $v\in V$. We ask for a TSP tour $T$ for $V$ that minimizes the expected length after cutting $T$ short by skipping the inactive customers. All known approximation algorithms select a nonempty subset $S$ of the customers and construct a master route solution, consisting of a TSP tour for $S$ and two edges connecting every customer $v\in V\setminus S$ to a nearest customer in $S$. We address the following questions. If we randomly sample the subset $S$, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.

cs.DS↗

arXiv2023

A Better-Than-1.6-Approximation for Prize-Collecting TSP

Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below $1.6$, where the guarantee is with respect to the natural linear programming relaxation of the problem. This improves upon the previous best-known approximation ratio of $1.774$. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by $(1+\sqrt{5})/2 \approx 1.618$, the golden ratio. With some additional technical care we further improve it to $1.599$. Furthermore, we show that for the path version of Prize-Collecting TSP (known as Prize-Collecting Stroll) our approach yields an approximation guarantee of 1.6662, improving upon the previous best-known guarantee of 1.926.

cs.DS↗

arXiv2023

An improved approximation guarantee for Prize-Collecting TSP

We present a new approximation algorithm for the (metric) prize-collecting traveling salesperson problem (PCTSP). In PCTSP, opposed to the classical traveling salesperson problem (TSP), one may not include a vertex of the input graph in the returned tour at the cost of a given vertex-dependent penalty, and the objective is to balance the length of the tour and the incurred penalties for omitted vertices by minimizing the sum of the two. We present an algorithm that achieves an approximation guarantee of $1.774$ with respect to the natural linear programming relaxation of the problem. This significantly reduces the gap between the approximability of classical TSP and PCTSP, beating the previously best known approximation factor of $1.915$. As a key ingredient of our improvement, we present a refined decomposition technique for solutions of the LP relaxation, and show how to leverage components of that decomposition as building blocks for our tours.

cs.DS↗

arXiv2022

Vehicle Routing with Time-Dependent Travel Times: Theory, Practice, and Benchmarks

We develop theoretical foundations and practical algorithms for vehicle routing with time-dependent travel times. We also provide new benchmark instances and experimental results. First, we study basic operations on piecewise linear arrival time functions. In particular, we devise a faster algorithm to compute the pointwise minimum of a set of piecewise linear functions and a monotonicity-preserving variant of the Imai-Iri algorithm to approximate an arrival time function with fewer breakpoints. Next, we show how to evaluate insertion and deletion operations in tours efficiently and update the underlying data structure faster than previously known when a tour changes. Evaluating a tour also requires a scheduling step which is non-trivial in the presence of time windows and time-dependent travel times. We show how to perform this in linear time. Based on these results, we develop a local search heuristic to solve real-world vehicle routing problems with various constraints efficiently and report experimental results on classical benchmarks. Since most of these do not have time-dependent travel times, we generate and publish new benchmark instances that are based on real-world data. This data also demonstrates the importance of considering time-dependent travel times in instances with tight time windows.

cs.DS↗

arXiv2020

Improving the Approximation Ratio for Capacitated Vehicle Routing

We devise a new approximation algorithm for capacitated vehicle routing. Our algorithm yields a better approximation ratio for general capacitated vehicle routing as well as for the unit-demand case and the splittable variant. Our results hold in arbitrary metric spaces. This is the first improvement upon the classical tour partitioning algorithm by Haimovich and Rinnooy Kan and Altinkemer and Gavish.

cs.DS↗