TY - JOUR

T1 - Probabilistic Analysis of Optimization Problems on Sparse Random Shortest Path Metrics

AU - Klootwijk, Stefan

AU - Manthey, Bodo

N1 - Funding Information:
An earlier version of this work has appeared in the Proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). This research was supported by NWO Grant 613.001.402.
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/12

Y1 - 2023/12

N2 - Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős–Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on sparse graphs with ‘fast growing cut sizes’, i.e. graphs for which | δ(U) | = Ω (| U| ε) for some constant ε∈ (0 , 1) for all subsets U of the vertices, where δ(U) is the set of edges connecting U to the remaining vertices. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a sparse graph with fast growing cut sizes, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

AB - Simple heuristics for (combinatorial) optimization problems often show a remarkable performance in practice. Worst-case analysis often falls short of explaining this performance. Because of this, “beyond worst-case analysis” of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many (combinatorial) optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained in recent years, where random shortest path metrics generated from dense graphs (either complete graphs or Erdős–Rényi random graphs) have been used so far. In this paper we extend these findings to sparse graphs, with a focus on sparse graphs with ‘fast growing cut sizes’, i.e. graphs for which | δ(U) | = Ω (| U| ε) for some constant ε∈ (0 , 1) for all subsets U of the vertices, where δ(U) is the set of edges connecting U to the remaining vertices. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances generated from a sparse graph with fast growing cut sizes, we prove that the greedy heuristic for the minimum distance maximum matching problem, and the nearest neighbor and insertion heuristics for the traveling salesman problem all achieve a constant expected approximation ratio. Additionally, for instances generated from an arbitrary sparse graph, we show that the 2-opt heuristic for the traveling salesman problem also achieves a constant expected approximation ratio.

KW - Approximation algorithms

KW - Average-case analysis

KW - First-passage percolation

KW - Probabilistic analysis

KW - Random shortest path metrics

KW - UT-Hybrid-D

UR - http://www.scopus.com/inward/record.url?scp=85169114659&partnerID=8YFLogxK

U2 - 10.1007/s00453-023-01167-3

DO - 10.1007/s00453-023-01167-3

M3 - Article

AN - SCOPUS:85169114659

SN - 0178-4617

VL - 85

SP - 3793

EP - 3815

JO - Algorithmica

JF - Algorithmica

IS - 12

ER -