104 research outputs found
Game saturation of intersecting families
We consider the following combinatorial game: two players, Fast and Slow,
claim -element subsets of alternately, one at each turn,
such that both players are allowed to pick sets that intersect all previously
claimed subsets. The game ends when there does not exist any unclaimed
-subset that meets all already claimed sets. The score of the game is the
number of sets claimed by the two players, the aim of Fast is to keep the score
as low as possible, while the aim of Slow is to postpone the game's end as long
as possible. The game saturation number is the score of the game when both
players play according to an optimal strategy. To be precise we have to
distinguish two cases depending on which player takes the first move. Let
and denote the score of
the saturation game when both players play according to an optimal strategy and
the game starts with Fast's or Slow's move, respectively. We prove that
holds
Hardness of submodular cost allocation : lattice matching and a simplex coloring conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f1, ... , fk: 2V -> R+ and the goal is to partition V into k sets A1, ..., Ak so as to minimize the total cost sumi = 1,k fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi + h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3
On Communication Complexity of Fixed Point Computation
Brouwer's fixed point theorem states that any continuous function from a
compact convex space to itself has a fixed point. Roughgarden and Weinstein
(FOCS 2016) initiated the study of fixed point computation in the two-player
communication model, where each player gets a function from to
, and their goal is to find an approximate fixed point of the
composition of the two functions. They left it as an open question to show a
lower bound of for the (randomized) communication complexity of
this problem, in the range of parameters which make it a total search problem.
We answer this question affirmatively.
Additionally, we introduce two natural fixed point problems in the two-player
communication model.
Each player is given a function from to ,
and their goal is to find an approximate fixed point of the concatenation of
the functions.
Each player is given a function from to , and
their goal is to find an approximate fixed point of the interpolation of the
functions.
We show a randomized communication complexity lower bound of
for these problems (for some constant approximation factor).
Finally, we initiate the study of finding a panchromatic simplex in a
Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the
two-player communication model: A triangulation of the -simplex is
publicly known and one player is given a set and a coloring
function from to , and the other player is given a set
and a coloring function from to ,
such that , and their goal is to find a panchromatic
simplex. We show a randomized communication complexity lower bound of
for the aforementioned problem as well (when is large)
Sperner\u27s Lemma, The Brouwer Fixed Point Theorem, the Kakutani Fixed Point Theorem, and Their Applications in Social Sciences
Can a cake be divided amongst people in such a manner that each individual is content with their share? In a game, is there a combination of strategies where no player is motivated to change their approach? Is there a price where the demand for goods is entirely met by the supply in the economy and there is no tendency for anything to change? In this paper, we will prove the existence of envy-free cake divisions, equilibrium game strategies and equilibrium prices in the economy, as well as discuss what brings them together under one heading.
This paper examines three important results in mathematics: Sperner’s lemma, the Brouwer fixed point theorem and the Kakutani fixed point theorem, as well as the interconnection between these theorems. Fixed point theorems are central results of topology that discuss existence of points in the domain of a continuous function that are mapped under the function to itself or to a set containing the point. The Kakutani fixed point theorem can be thought of as a generalization of the Brouwer fixed point theorem. Sperner’s lemma, on the other hand, is often described as a combinatorial analog of the Brouwer fixed point theorem, if the assumptions of the lemma are developed as a function. In this thesis, we first introduce Sperner’s lemma and it serves as a building block for the proof of the fixed point theorem which in turn is used to prove the Kakutani fixed point theorem that is at the top of the pyramid.
This paper highlights the interdependence of the results and how they all are applicable to prove the existence of equilibria in fair division problems, game theory and exchange economies. Equilibrium means a state of rest, a point where opposing forces balance. Sperner’s lemma is applied to the cake cutting dilemma to find a division where no individual vies for another person’s share, the Brouwer fixed point theorem is used to prove the existence of an equilibrium game strategy where no player is motivated to change their approach, and the Kakutani fixed point theorem proves that there exists a price where the demand for goods is completely met by the supply and there is no tendency for prices to change within the market
KKM type theorems with boundary conditions
We consider generalizations of Gale's colored KKM lemma and Shapley's KKMS
theorem. It is shown that spaces and covers can be much more general and the
boundary KKM rules can be substituted by more weaker boundary assumptions.Comment: 13 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1406.6672 by other author
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
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