skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Why Is Maximum Clique Often Easy in Practice?
To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with 1,000 vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many naturally occurring networks? In this paper, we provide an explanation. First, we observe that the graph’s clique number ω is very near to the graph’s degeneracy d in most real-life instances. This observation motivates a main contribution of this paper, which is an algorithm for the maximum clique problem that runs in time polynomial in the size of the graph, but exponential in the gap [Formula: see text] between the clique number ω and its degeneracy-based upper bound d+1. When this gap [Formula: see text] can be treated as a constant, as is often the case for real-life graphs, the proposed algorithm runs in time [Formula: see text]. This provides a rigorous explanation for the apparent easiness of these instances despite the intractability of the problem in the worst case. Further, our implementation of the proposed algorithm is actually practical—competitive with the best approaches from the literature.  more » « less
Award ID(s):
1662757
PAR ID:
10250666
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Operations Research
Volume:
68
Issue:
6
ISSN:
0030-364X
Page Range / eLocation ID:
1866 to 1895
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Mathematics of Operations Research)
    We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors). 
    more » « less
  2. ; ; ; (, International Journal of Computational Geometry & Applications)
    Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let [Formula: see text] and [Formula: see text] be two polygonal curves in [Formula: see text] with [Formula: see text] and [Formula: see text] vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in [Formula: see text] time, (3) a linear-time [Formula: see text]-approximation algorithm, and (4) a data structure that supports [Formula: see text]-time decision queries, where [Formula: see text] is the number of vertices of the query curve and [Formula: see text] the number of vertices of the preprocessed curve. 
    more » « less
  3. ; (, International Journal of Computational Geometry & Applications)
    We study the problem of covering barrier points by mobile sensors. Each sensor is represented by a point in the plane with the same covering range [Formula: see text] so that any point within distance [Formula: see text] from the sensor can be covered by the sensor. Given a set [Formula: see text] of [Formula: see text] points (called “barrier points”) and a set [Formula: see text] of [Formula: see text] points (representing the “sensors”) in the plane, the problem is to move the sensors so that each barrier point is covered by at least one sensor and the maximum movement of all sensors is minimized. The problem is NP-hard. In this paper, we consider two line-constrained variations of the problem and present efficient algorithms that improve the previous work. In the first problem, all sensors are given on a line [Formula: see text] and are required to move on [Formula: see text] only while the barrier points can be anywhere in the plane. We propose an [Formula: see text] time algorithm for the problem. We also consider the weighted case where each sensor has a weight; we give an [Formula: see text] time algorithm for this case. In the second problem, all barrier points are on [Formula: see text] while all sensors are in the plane but are required to move onto [Formula: see text] to cover all barrier points. We also solve the weighted case in [Formula: see text] time. 
    more » « less
  4. ; ; (, INFORMS Journal on Optimization)
    Motivated by the importance of user engagement as a crucial element in cascading leaving of users from a social network, we study identifying a largest relaxed variant of a degree-based cohesive subgraph: the maximum anchored k-core problem. Given graph [Formula: see text] and integers k and b, the maximum anchored k-core problem seeks to find a largest subset of vertices [Formula: see text] that induces a subgraph with at least [Formula: see text] vertices of degree at least k. We introduce a new integer programming (IP) formulation for the maximum anchored k-core problem and conduct a polyhedral study on the polytope of the problem. We show the linear programming relaxation of the proposed IP model is at least as strong as that of a naïve formulation. We also identify facet-defining inequalities of the IP formulation. Furthermore, we develop inequalities and fixing procedures to improve the computational performance of our IP model. We use benchmark instances to compare the computational performance of the IP model with (i) the naïve IP formulation and (ii) two existing heuristic algorithms. Our proposed IP model can optimally solve half of the benchmark instances that cannot be solved to optimality either by the naïve model or the existing heuristic approaches. Funding: This work is funded by the National Science Foundation (NSF) [Grant DMS-2318790] titled AMPS: Novel Combinatorial Optimization Techniques for Smartgrids and Power Networks. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0024 . 
    more » « less
  5. ; ; (, Asia-Pacific Journal of Operational Research)
    Let [Formula: see text] be a directed graph associated with a weight [Formula: see text]. For an edge-cut [Formula: see text] of [Formula: see text], the average weight of [Formula: see text] is denoted and defined as [Formula: see text]. An optimal edge-cut with average weight is an edge-cut [Formula: see text] such that [Formula: see text] is maximum among all edge-cuts (or minimum, symmetrically). In this paper, a polynomial algorithm for this problem is proposed for finding an optimal edge-cut in a rooted tree separating the root and the set of all leafs. This algorithm enables us to develop an automatic clustering method with more accurate detection of community output. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email