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

The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of ~O(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] presented an algorithm that computes in ~O(m√ n + n2) time an estimate D for the diameter D, such that ⌊ 2/3 D ⌋ ≤ ^D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years.Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et. al, producing an algorithm with the same estimate but with an expected running time of ~O(m√ n). We thus show that for all sparse enough graphs, the diameter can be 3/2-approximated in o(n2) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node.We also provide strong evidence that our diameter approximation result may be hard to improve. We show that if for some constant e>0 there is an O(m2-e) time (3/2-e)-approximation algorithm for the diameter of undirected unweighted graphs, then there is an O*( (2-δ)n) time algorithm for CNF-SAT on n variables for constant δ>0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false.Motivated by this negative result, we give several improved diameter approximation algorithms for special cases. We show for instance that for unweighted graphs of constant diameter D not divisible by 3, there is an O(m2-e) time algorithm that gives a (3/2-e) approximation for constant e>0. This is interesting since the diameter approximation problem is hardest to solve for small D.

/pdf/fast-approximation-algorithms-for-the-diameter-and-radius-of-87jak6ngfp.pdf

Fast approximation algorithms for the diameter and radius of sparse graphs

Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn$^{\rm 1/{\it k}}$) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn$^{\rm 1+1/{\it k}}$) capable of returning in O(k) time an approximation $\hat{\delta}(u,v)$ of the distance δ(u,v) from u to v in G that satisfies $\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)$, for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn$^{\rm 1+1/{\it k}}$log n). We present here a deterministic O(kmn$^{\rm 1/{\it k}}$) time algorithm for constructing oracles of size O(kn$^{\rm 1+1/{\it k}}$). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor.

Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn$^{\rm 1+1/{\it k}}$) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

Deterministic constructions of approximate distance oracles and spanners

We obtain the following results related to dynamic versions of the shortest-paths problem: 
Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems.
A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of $\tilde {O}(m\sqrt{n})$ (we use $\tilde {O}$ to hide small poly-logarithmic factors) and a worst case query time is O(n3/4).
A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

/pdf/on-dynamic-shortest-paths-problems-iwftxp8rpq.pdf

On Dynamic Shortest Paths Problems

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n0, there exists a graph built by n ≥ n0 players which contains cycles and forms a non-transient Nash equilibrium, for any α with 1

/pdf/on-nash-equilibria-for-a-network-creation-game-3cuoumfedc.pdf

On nash equilibria for a network creation game

We obtain the following results related to dynamic versions of the shortest-paths problem: (i) Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. (ii) A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of O(mn) and a worst case query time is O(n 3/4 ). (iii) A deterministic O(n 2 log n) time algorithm for constructing a (log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.

Liam Roditty

Papers

Fast approximation algorithms for the diameter and radius of sparse graphs

Deterministic constructions of approximate distance oracles and spanners

On Dynamic Shortest Paths Problems

On nash equilibria for a network creation game

On Dynamic Shortest Paths Problems