18 research outputs found
Reallocating Multiple Facilities on the Line
We study the multistage -facility reallocation problem on the real line,
where we maintain facility locations over stages, based on the
stage-dependent locations of agents. Each agent is connected to the nearest
facility at each stage, and the facilities may move from one stage to another,
to accommodate different agent locations. The objective is to minimize the
connection cost of the agents plus the total moving cost of the facilities,
over all stages. -facility reallocation was introduced by de Keijzer and
Wojtczak, where they mostly focused on the special case of a single facility.
Using an LP-based approach, we present a polynomial time algorithm that
computes the optimal solution for any number of facilities. We also consider
online -facility reallocation, where the algorithm becomes aware of agent
locations in a stage-by-stage fashion. By exploiting an interesting connection
to the classical -server problem, we present a constant-competitive
algorithm for facilities
Evolutionary Game Theory Squared: Evolving Agents in Endogenously Evolving Zero-Sum Games
The predominant paradigm in evolutionary game theory and more generally
online learning in games is based on a clear distinction between a population
of dynamic agents that interact given a fixed, static game. In this paper, we
move away from the artificial divide between dynamic agents and static games,
to introduce and analyze a large class of competitive settings where both the
agents and the games they play evolve strategically over time. We focus on
arguably the most archetypal game-theoretic setting -- zero-sum games (as well
as network generalizations) -- and the most studied evolutionary learning
dynamic -- replicator, the continuous-time analogue of multiplicative weights.
Populations of agents compete against each other in a zero-sum competition that
itself evolves adversarially to the current population mixture. Remarkably,
despite the chaotic coevolution of agents and games, we prove that the system
exhibits a number of regularities. First, the system has conservation laws of
an information-theoretic flavor that couple the behavior of all agents and
games. Secondly, the system is Poincar\'{e} recurrent, with effectively all
possible initializations of agents and games lying on recurrent orbits that
come arbitrarily close to their initial conditions infinitely often. Thirdly,
the time-average agent behavior and utility converge to the Nash equilibrium
values of the time-average game. Finally, we provide a polynomial time
algorithm to efficiently predict this time-average behavior for any such
coevolving network game.Comment: To appear in AAAI 202
Semi Bandit Dynamics in Congestion Games: Convergence to Nash Equilibrium and No-Regret Guarantees
In this work, we introduce a new variant of online gradient descent, which
provably converges to Nash Equilibria and simultaneously attains sublinear
regret for the class of congestion games in the semi-bandit feedback setting.
Our proposed method admits convergence rates depending only polynomially on the
number of players and the number of facilities, but not on the size of the
action set, which can be exponentially large in terms of the number of
facilities. Moreover, the running time of our method has polynomial-time
dependence on the implicit description of the game. As a result, our work
answers an open question from (Du et. al, 2022).Comment: ICML 202
On the Approximability of Multistage Min-Sum Set Cover
We investigate the polynomial-time approximability of the multistage version
of Min-Sum Set Cover (), a natural and intriguing generalization
of the classical List Update problem. In , we maintain a
sequence of permutations on elements, based
on a sequence of requests . We aim to minimize the total
cost of updating to , quantified by the Kendall tau
distance , plus the total cost of
covering each request with the current permutation , quantified by
the position of the first element of in .
Using a reduction from Set Cover, we show that does not admit
an -approximation, unless , and that any
(resp. ) approximation to implies a
sublogarithmic (resp. ) approximation to Set Cover (resp. where each
element appears at most times). Our main technical contribution is to show
that can be approximated in polynomial-time within a factor of
in general instances, by randomized rounding, and within a factor
of , if all requests have cardinality at most , by deterministic
rounding
Node-Max-Cut and the Complexity of Equilibrium in Linear Weighted Congestion Games
In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkable gap regarding the complexity of PNE in congestion games with weighted and unweighted players, since in case of unweighted players, a PNE can be easily computed by either a simple greedy algorithm (for series-parallel networks) or any better response dynamics (when the latency is equal to the congestion). For the latter of the results above, we need to show first that computing a local optimum of a natural restriction of Max-Cut, which we call Node-Max-Cut, is PLS-complete. In Node-Max-Cut, the input graph is vertex-weighted and the weight of each edge is equal to the product of the weights of its endpoints. Due to the very restricted nature of Node-Max-Cut, the reduction requires a careful combination of new gadgets with ideas and techniques from previous work. We also show how to compute efficiently a (1+?)-approximate equilibrium for Node-Max-Cut, if the number of different vertex weights is constant
Maximum Independent Set: Self-Training through Dynamic Programming
This work presents a graph neural network (GNN) framework for solving the
maximum independent set (MIS) problem, inspired by dynamic programming (DP).
Specifically, given a graph, we propose a DP-like recursive algorithm based on
GNNs that firstly constructs two smaller sub-graphs, predicts the one with the
larger MIS, and then uses it in the next recursive call. To train our
algorithm, we require annotated comparisons of different graphs concerning
their MIS size. Annotating the comparisons with the output of our algorithm
leads to a self-training process that results in more accurate self-annotation
of the comparisons and vice versa. We provide numerical evidence showing the
superiority of our method vs prior methods in multiple synthetic and real-world
datasets.Comment: Accepted in NeurIPS 202