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

Let $$S \subset \mathbb {R}^2$$
 be a set of n sites. The unit disk graph $${{\mathrm{UD}}}(S)$$
 on S has vertex set S and an edge between two distinct sites $$s,t \in S$$
 if and only if s and t have Euclidean distance $$|st| \le 1$$
 . A routing scheme R for $${{\mathrm{UD}}}(S)$$
 assigns to each site $$s \in S$$
 a label $$\ell (s)$$
 and a routing table $$\rho (s)$$
 . For any two sites $$s, t \in S$$
 , the scheme R must be able to route a packet from s to t in the following way: given a current site r (initially, $$r = s$$
 ), a header h (initially empty), and the label $$\ell (t)$$
 of the target, the scheme R consults the routing table $$\rho (r)$$
 to compute a neighbor $$r'$$
 of r, a new header $$h'$$
 , and the label $$\ell (t')$$
 of an intermediate target $$t'$$
 . (The label of the original target may be stored at the header $$h'$$
 .) The packet is then routed to $$r'$$
 , and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in $${{\mathrm{UD}}}(S)$$
 , over all pairs of distinct sites in S. For any given $$\varepsilon > 0$$
 , we show how to construct a routing scheme for $${{\mathrm{UD}}}(S)$$
 with stretch $$1+\varepsilon $$
 using labels of $$O(\log n)$$
 bits and routing tables of $$O(\varepsilon ^{-5}\log ^2 n \log ^2 D)$$
 bits, where D is the (Euclidean) diameter of $${{\mathrm{UD}}}(S)$$
 . The header size is $$O(\log n \log D)$$
 bits.

Routing in Unit Disk Graphs

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. 
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: 
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. 
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly $(3/2+\epsilon)$-approximation to Diameter in directed or undirected graphs can be maintained decrementally in total time $m^{1+o(1)}\sqrt{n}/\epsilon^2$. This nearly matches the static $3/2$-approximation algorithm for the problem that is known to be conditionally optimal.

Algorithms and Hardness for Diameter in Dynamic Graphs

Let V be a set of points in a d-dimensional lp-metric space. Let s, t e V and let L be any real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set is at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s-t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and any real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper present first an algorithm for the apBLSP problem in any lp-metric which, for any fixed e > 0, computes in O(n3n = log2 n · e-d)) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1 + e). It requires O(n2log n) space and distance queries are answered in O (log log n) time. This improves on an algorithm with running time of O(n5) given by Bose et al. in [8]. We present also an algorithm for the stBLSP problem that, given s, t ∈ V and a real number L, computes in O(n · polylog(n)) the exact L-bounded shortest path from s to t. This algorithm works in l1 and l∞ metrics. In the Euclidean metric we also obtain an exact algorithm but with a running time of O(n4/3+e), for any e > 0. We end by showing that for any weighted directed graph there is a data structure of size O(n2.5log n) which is capable of answering path queries with a multiplicative error of (1 + e) in O (log log n + e) time, where e is the length of the reported path.Our results improve upon the results given by Bose et al. [8]. Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of an independent interest.

On bounded leg shortest paths problems

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: 
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. 
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.

https://drops.dagstuhl.de/opus/volltexte/2019/10589/pdf/LIPIcs-ICALP-2019-13.pdf/

Algorithms and Hardness for Diameter in Dynamic Graphs.

We present the first approximation algorithm for finding the $k$ shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of $mn$. It is deterministic and has a running time of $O(k(m\sqrt{n}+n^{3/2}\log n))$, where $m$ is the number of edges in the graph and $n$ is the number of vertices. Let $s,t\in V$; the length of the $i$th simple path from $s$ to $t$ computed by our algorithm is at most $\frac{3}{2}$ times the length of the $i$th shortest simple path from $s$ to $t$. The best algorithms for computing the exact $k$ shortest simple paths connecting a pair of vertices in a weighted directed graph are due to Yen [Management Sci., 17 (1970/1971), pp. 712-716] and Lawler [Management Sci., 18 (1971/1972), pp. 401-405]. The running time of their algorithms, using modern data structures, is $O(k(mn+n^2\log n))$. Both algorithms are from the early 70s. Although this problem and other variants of the $k$ shortest path problem has drawn a lot of attention during the last three and a half decades, the $O(k(mn+n^2\log n))$ bound is still unbeaten.

Liam Roditty

Papers

Routing in Unit Disk Graphs

Algorithms and Hardness for Diameter in Dynamic Graphs

On bounded leg shortest paths problems

Algorithms and Hardness for Diameter in Dynamic Graphs.

On the $k$ Shortest Simple Paths Problem in Weighted Directed Graphs