Dimitris Fotakis

The Structure and Complexity of Nash Equilibria for a Selfish Routing Game

In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models selfish routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.We embark on a systematic study of several algorithmic problems related to the computation of Nash equilibria for the selfish routing game we consider. In a nutshell, these problems relate to deciding the existence of a Nash equilibrium, constructing a Nash equilibrium with given support characteristics, constructing the worst Nash equilibrium (the one with maximum social cost), constructing the best Nash equilibrium (the one with minimum social cost), or computing the social cost of a (given) Nash equilibrium. Our work provides a comprehensive collection of efficient algorithms, hardness results (both as NP-hardness and #P-completeness results), and structural results for these algorithmic problems. Our results span and contrast a wide range of assumptions on the syntax of the Nash equilibria and on the parameters of the system.

Space Efficient Hash Tables With Worst Case Constant Access Time

Abstract We generalize Cuckoo Hashing to d-ary Cuckoo Hashing and show how this yields a simple hash table data structure that stores n elements in (1 + ε)n memory cells, for any constant ε > 0. Assuming uniform hashing, accessing or deleting table entries takes at most d=O (ln (1/ε)) probes and the expected amortized insertion time is constant. This is the first dictionary that has worst case constant access time and expected constant update time, works with (1 + ε)n space, and supports satellite information. Experiments indicate that d = 4 probes suffice for ε ≈ 0.03. We also describe variants of the data structure that allow the use of hash functions that can be evaluated in constant time.

Atomic congestion games among coalitions

We consider algorithmic questions concerning the existence, tractability, and quality of Nash equilibria, in atomic congestion games among users participating in selfish coalitions. We introduce a coalitional congestion model among atomic players and demonstrate many interesting similarities with the noncooperative case. For example, there exists a potential function proving the existence of pure Nash equilibria (PNE) in the unrelated parallel links setting; in the network setting, the finite improvement property collapses as soon as we depart from linear delays, but there is an exact potential (and thus PNE) for linear delays. The price of anarchy on identical parallel links demonstrates a quite surprising threshold behavior: It persists on being asymptotically equal to that in the case of the noncooperative KP-model, unless the number of coalitions is sublogarithmic. We also show crucial differences, mainly concerning the hardness of algorithmic problems that are solved efficiently in the noncooperative case. Although we demonstrate convergence to robust PNE, we also prove the hardness of computing them. On the other hand, we propose a generalized fully mixed Nash equilibrium that can be efficiently constructed in most cases. Finally, we propose a natural improvement policy and prove its convergence in pseudopolynomial time to PNE which are robust against (even dynamically forming) coalitions of small size.

Enumerating subgraph instances using map-reduce

The theme of this paper is how to find all instances of a given “sample” graph in a larger “data graph,” using a single round of map-reduce. For the simplest sample graph, the triangle, we improve upon the best known such algorithm. We then examine the general case, considering both the communication cost between mappers and reducers and the total computation cost at the reducers. To minimize communication cost, we exploit the techniques of [1] for computing multiway joins (evaluating conjunctive queries) in a single map-reduce round. Several methods are shown for translating sample graphs into a union of conjunctive queries with as few queries as possible. We also address the matter of optimizing computation cost. Many serial algorithms are shown to be “convertible,” in the sense that it is possible to partition the data graph, explore each partition in a separate reducer, and have the total computation cost at the reducers be of the same order as the computation cost of the serial algorithm.

On the Competitive Ratio for Online Facility Location

Abstract We consider the problem of Online Facility Location, where the demand points arrive online and must be assigned irrevocably to an open facility upon arrival. The objective is to minimize the sum of facility and assignment costs. We prove that the competitive ratio for Online Facility Location is Θ

Symmetry in network congestion games: pure equilibria and anarchy cost

(\frac{\log n}{\log\log n})

The structure and complexity of Nash equilibria for a selfish routing game

. On the negative side, we show that no randomized algorithm can achieve a competitive ratio better than Ω

A primal-dual algorithm for online non-uniform facility location

(\frac{\log n}{\log\log n})

On the Power of Deterministic Mechanisms for Facility Location Games

against an oblivious adversary even if the demands lie on a line segment. On the positive side, we present a deterministic algorithm which achieves a competitive ratio of

NP-Completeness Results and Efficient Approximations for Radiocoloring in Planar Graphs

\mathrm{O}(\frac{\log n}{\log\log n})