12 research outputs found
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
A Geometric Approach to Combinatorial Fixed-Point Theorems
We develop a geometric framework that unifies several different combinatorial
fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing
them to be different geometric manifestations of the same topological
phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like
fixed-point theorems involving an exponential-sized label set; (2) a
generalization of Fan's parity proof of Tucker's Lemma to a much broader class
of label sets; and (3) direct proofs of several Sperner-like lemmas from
Tucker's lemma via explicit geometric embeddings, without the need for
topological fixed-point theorems. Our work naturally suggests several
interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201
A Low-Complexity Approach to Distributed Cooperative Caching with Geographic Constraints
We consider caching in cellular networks in which each base station is
equipped with a cache that can store a limited number of files. The popularity
of the files is known and the goal is to place files in the caches such that
the probability that a user at an arbitrary location in the plane will find the
file that she requires in one of the covering caches is maximized.
We develop distributed asynchronous algorithms for deciding which contents to
store in which cache. Such cooperative algorithms require communication only
between caches with overlapping coverage areas and can operate in asynchronous
manner. The development of the algorithms is principally based on an
observation that the problem can be viewed as a potential game. Our basic
algorithm is derived from the best response dynamics. We demonstrate that the
complexity of each best response step is independent of the number of files,
linear in the cache capacity and linear in the maximum number of base stations
that cover a certain area. Then, we show that the overall algorithm complexity
for a discrete cache placement is polynomial in both network size and catalog
size. In practical examples, the algorithm converges in just a few iterations.
Also, in most cases of interest, the basic algorithm finds the best Nash
equilibrium corresponding to the global optimum. We provide two extensions of
our basic algorithm based on stochastic and deterministic simulated annealing
which find the global optimum.
Finally, we demonstrate the hit probability evolution on real and synthetic
networks numerically and show that our distributed caching algorithm performs
significantly better than storing the most popular content, probabilistic
content placement policy and Multi-LRU caching policies.Comment: 24 pages, 9 figures, presented at SIGMETRICS'1
Clearing Financial Networks with Derivatives: From Intractability to Algorithms
Financial networks raise a significant computational challenge in identifying
insolvent firms and evaluating their exposure to systemic risk. This task,
known as the clearing problem, is computationally tractable when dealing with
simple debt contracts. However under the presence of certain derivatives called
credit default swaps (CDSes) the clearing problem is -complete.
Existing techniques only show -hardness for finding an
-solution for the clearing problem with CDSes within an unspecified
small range for .
We present significant progress in both facets of the clearing problem: (i)
intractability of approximate solutions; (ii) algorithms and heuristics for
computable solutions. Leveraging (FOCS'22), we provide
the first explicit inapproximability bound for the clearing problem involving
CDSes. Our primal contribution is a reduction from
which establishes that finding approximate solutions is -hard
within a range of roughly 5%.
To alleviate the complexity of the clearing problem, we identify two
meaningful restrictions of the class of financial networks motivated by
regulations: (i) the presence of a central clearing authority; and (ii) the
restriction to covered CDSes. We provide the following results: (i.) The
-hardness of approximation persists when central clearing
authorities are introduced; (ii.) An optimisation-based method for solving the
clearing problem with central clearing authorities; (iii.) A polynomial-time
algorithm when the two restrictions hold simultaneously
Nash equilibria, gale strings, and perfect matchings
This thesis concerns the problem 2-NASH of ļ¬nding a Nash equilibrium of a bimatrix
game, for the special class of so-called āhard-to-solveā bimatrix games. The term āhardto-solveā relates to the exponential running time of the famous and often used Lemkeā
Howson algorithm for this class of games. The games are constructed with the help of
dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled
bitstrings deļ¬ned by the āGale evenness conditionā that characterise the vertices of these
polytopes.
We deļ¬ne the combinatorial problem āAnother completely labeled Gale stringā whose
solutions deļ¬ne the Nash equilibria of any game deļ¬ned by cyclic polytopes, including
the games where the LemkeāHowson algorithm takes exponential time. We show that
āAnother completely labeled Gale stringā is solvable in polynomial time by a reduction to
the āPerfect matchingā problem in Euler graphs. We adapt the LemkeāHowson algorithm
to pivot from one perfect matching to another and show that again for a certain class
of graphs this leads to exponential behaviour. Furthermore, we prove that completely
labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the
LemkeāHowson algorithm connects two strings or matchings of opposite signs.
The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that
counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in ļ¬nding an algorithm that computes
a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difļ¬culties concerning a general
polynomial-time algorithm for this problem by means of negative results that demonstrate
why a number of approaches towards such an algorithm are unlikely to be successful
Recent Advances in Cellular D2D Communications
Device-to-device (D2D) communications have attracted a great deal of attention from researchers in recent years. It is a promising technique for offloading local traffic from cellular base stations by allowing local devices, in physical proximity, to communicate directly with each other. Furthermore, through relaying, D2D is also a promising approach to enhancing service coverage at cell edges or in black spots. However, there are many challenges to realizing the full benefits of D2D. For one, minimizing the interference between legacy cellular and D2D users operating in underlay mode is still an active research issue. With the 5th generation (5G) communication systems expected to be the main data carrier for the Internet-of-Things (IoT) paradigm, the potential role of D2D and its scalability to support massive IoT devices and their machine-centric (as opposed to human-centric) communications need to be investigated. New challenges have also arisen from new enabling technologies for D2D communications, such as non-orthogonal multiple access (NOMA) and blockchain technologies, which call for new solutions to be proposed. This edited book presents a collection of ten chapters, including one review and nine original research works on addressing many of the aforementioned challenges and beyond