We study the multilevel Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals T require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals u, v with priorities P(u), P(v) are connected using edges of rate min{P(u),P(v)} or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of nonproportional costs, this problem is hard to approximate with ratio c log log n, where n is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min{2(ln T  + 1), lρ}approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and l is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ ≈ 1.39, for example). In this paper, we describe a natural generalization to the multilevel case of the classical (singlelevel) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. Wemore »
This content will become publicly available on June 30, 2023
LearningAugmented Algorithms for Online Steiner Tree
This paper considers the recently popular beyondworstcase algorithm analysis model which integrates machinelearned predictions with online algorithm design. We consider the online Steiner tree problem in this model for both directed and undirected graphs. Steiner tree is known to have strong lower bounds in the online setting and any algorithm’s worstcase guarantee is far from desirable. This paper considers algorithms that predict which terminal arrives online. The predictions may be incorrect and the algorithms’ performance is parameterized by the number of incorrectly predicted terminals. These guarantees ensure that algorithms break through the online lower bounds with good predictions and the competitive ratio gracefully degrades as the prediction error grows. We then observe that the theory is predictive of what will occur empirically. We show on graphs where terminals are drawn from a distribution, the new online algorithms have strong performance even with modestly correct predictions.
 Publication Date:
 NSFPAR ID:
 10351515
 Journal Name:
 Proceedings of the AAAI Conference on Artificial Intelligence
 Volume:
 36
 Issue:
 8
 Page Range or eLocationID:
 8744 to 8752
 ISSN:
 21595399
 Sponsoring Org:
 National Science Foundation
More Like this


Statistics of small subgraph counts such as triangles, fourcycles, and st paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most of these problems become hard in the dynamic setting when considering the worst case. In this paper, we ask whether the question of small subgraph counting over dynamic graphs is hard also in the average case. We consider the simplest possible average case model where the updates follow an ErdősRényi graph: each update selects a pair of vertices (u, v) uniformly at random and flips the existence of the edge (u, v). We develop new lower bounds and matching algorithms in this model for counting fourcycles, counting triangles through a specified point s, or a random queried point, and st paths of length 3, 4 and 5. Our results indicate while computing st paths of length 3, and 4 are easy in the average case with O(1) update time (note that they are hard in the worst case), it becomes hard when considering st paths of length 5. We introducemore »

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimumcost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasipolynomial time O(log²k/log log k)approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general DegreeBounded Directed Steiner Tree (DBDST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasipolynomial time (O(log n log k), O(log² n))bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first nontrivial result for the problem. While our costguarantee is nearly optimal, the degree violation factor of O(log²n) is an O(logmore »

Miller, Avery (Ed.)In this paper, we consider contention resolution algorithms that are augmented with predictions about the network. We begin by studying the natural setup in which the algorithm is provided a distribution defined over the possible network sizes that predicts the likelihood of each size occurring. The goal is to leverage the predictive power of this distribution to improve on worstcase time complexity bounds. Using a novel connection between contention resolution and information theory, we prove lower bounds on the expected time complexity with respect to the Shannon entropy of the corresponding network size random variable, for both the collision detection and no collision detection assumptions. We then analyze upper bounds for these settings, assuming now that the distribution provided as input might differ from the actual distribution generating network sizes. We express their performance with respect to both entropy and the statistical divergence between the two distributionsallowing us to quantify the cost of poor predictions. Finally, we turn our attention to the related perfect advice setting, parameterized with a length b ≥ 0, in which all active processes in a given execution are provided the best possible b bits of information about their network. We provide tight bounds on themore »

We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparisonbased (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT1 Congest model if noncomparisonbased algorithms are permitted. Anmore »