Computational geometry

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is $O(\log k)$, where $k$ is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-89te3x19ow.pdf

The Price of Stability for Network Design with Fair Cost Allocation

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

Journal of the ACM

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-4b3bbv50mq.pdf

The price of stability for network design with fair cost allocation

Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. [SODA 2014] showed that the diameter can be approximated within a multiplicative factor of 3/2 in O(m3/2) time. Furthermore, Roditty and Vassilevska W. [STOC 13] showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no O(n2−e) time algorithm can achieve an approximation factor better than 3/2 in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than 3/2. It was, however, completely plausible that a 3/2-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no O(m3/2−e) time algorithm can achieve an approximation factor better than 5/3. Another fundamental set of graph parameters are the Eccentricities. The Eccentricity of a vertex v is the distance between v and the farthest vertex from v. Chechik et al. [SODA 2014] showed that the Eccentricities of all vertices can be approximated within a factor of 5/3 in O(m3/2) time and Abboud et al. [SODA 2016] showed that no O(n2−e) algorithm can achieve better than 5/3 approximation in sparse graphs. We show that the runtime of the 5/3 approximation algorithm is also optimal by proving that under SETH, there is no O(m3/2−e) algorithm that achieves a better than 9/5 approximation. We also show that no near-linear time algorithm can achieve a better than 2 approximation for the Eccentricities. This is the first lower bound in fine-grained complexity that addresses near-linear time computation. We show that our lower bound for near-linear time algorithms is essentially tight by giving an algorithm that approximates Eccentricities within a 2+δ factor in O(m/δ) time for any 0 To establish the above lower bounds we study the S-T Diameter problem: Given a graph and two subsets S and T of vertices, output the largest distance between a vertex in S and a vertex in T. We give new algorithms and show tight lower bounds that serve as a starting point for all other hardness results. Our lower bounds apply only to sparse graphs. We show that for dense graphs, there are near-linear time algorithms for S-T Diameter, Diameter and Eccentricities, with almost the same approximation guarantees as their O(m3/2) counterparts, improving upon the best known algorithms for dense graphs.

https://dl.acm.org/doi/pdf/10.1145/3188745.3188950

Towards tight approximation bounds for graph diameter and eccentricities

We obtain the first approximation algorithm for finding the k-simple shortest paths connecting a pair of vertices in a weighted directed graph. Our algorithm is deterministic and has a running time of O(k(m√n + n3/2 log n)) where m is the number of edges in the graph and n is the number of vertices. Let s, t e V; the length of the i-th simple path from s to t computed by our algorithm is at most 3/2 times the length of the i-th shortest simple path from s to t. The best algorithms for computing the exact k-simple shortest paths connecting a pair of vertices in a weighted directed graph are due to Yen [19] and Lawler [13]. The running time of their algorithms, using modern data structures, is O(k(mn + n2 log n)). Both algorithms are from the early 70's. Although this problem and other variants of the k-shortest path problem drew a lot of attention during the last three and a half decades, the O(k(mn + n2 log n)) bound is still unbeaten.

On the K-simple shortest paths problem in weighted directed graphs

Given a weighted undirected graph, our basic goal is to represent all pair wise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse with $m=\wt O(n)$ edges. In this paper, for any fixed positive integers $k$ and $\ell$, we obtain stretches $\alpha=2k+1\pm\frac{2}{\ell}= 2k+1-\frac{2}{\ell}, 2k+1+\frac{2}{\ell}$, using space $S(\alpha, m) = \wt O(m^{1+2/(\alpha+1)})$. The query time is $O(k+\ell)=O(1)$. For integer stretches, this coincides with the previous bounds (odd stretches with $\ell=1$ and even stretches with $\ell=2$). The infinity of fractional stretches between consecutive integers are all new (even though $\ell$ is fixed as a constant independent of the input, the number of integers $\ell$ is still countably infinite). % We will argue that the new fractional points are not just arbitrary, but that they, at least for fixed stretches below $3$, provide a complete picture of the inherent trade-off between stretch and space in $m$. Consider any fixed stretch $\alpha

/pdf/a-new-infinity-of-distance-oracles-for-sparse-graphs-4ogeyeb2uv.pdf

A New Infinity of Distance Oracles for Sparse Graphs

We obtain a new fully dynamic algorithm for maintaining the transitive closure of a directed graph. Our algorithm maintains the transitive closure matrix in a total running time of O(mn p (ins p del) · n2), where ins (del) is the number of insert (delete) operations performed. Here n is the number of vertices in the graph and m is the initial number of edges in the graph. Obviously, reachability queries can be answered in constant time. The algorithm uses only O(n2) time which is essentially optimal for maintaining the transitive closure matrix. Our algorithm can also support path queries. If v is reachable from u, the algorithm can produce a path from u to v in time proportional to the length of the path. The best previously known algorithm for the problem is due to Demetrescu and Italiano [2000]. Their algorithm has a total running time of O(n3 p (ins p del) · n2). The query time is also constant. In addition, we also present a simple algorithm for directed acyclic graphs (DAGs) with a total running time of O(mn p ins · n2 p del). Our algorithms are obtained by combining some new ideas with techniques of Italiano [1986, 1988], King [1999], King and Thorup [2001] and Frigioni et al. [2001]. We also note that our algorithms are extremely simple and can be easily implemented.

A faster and simpler fully dynamic transitive closure

In this paper we present an algorithm for maintaining a spanner over a dynamic set of points in constant doubling dimension metric spaces. For a set S of points in ℝ d , a t-spanner is a sparse graph on the points of S such that there is a path in the spanner between any pair of points whose total length is at most t times the distance between the points. We present the first fully dynamic algorithm for maintaining a spanner whose update time depends solely on the number of points in S. In particular, we show how to maintain a (1+e)-spanner with O(n/e d ) edges, where points can be inserted to S in an amortized update time of O(log n) and deleted from S in an amortized update time of $\tilde{O}(n^{1/3})$.

As a by-product of our techniques we obtain a simple incremental algorithm for constructing a (1+e)-spanner with O(n/e d ) edges in constant doubling dimension metric spaces whose running time is O(nlog n).

Liam Roditty

Papers

Towards tight approximation bounds for graph diameter and eccentricities

On the K-simple shortest paths problem in weighted directed graphs

A New Infinity of Distance Oracles for Sparse Graphs

A faster and simpler fully dynamic transitive closure

Fully Dynamic Geometric Spanners