15 research outputs found
Beyond Worst-Case (In)approximability of Nonsubmodular Influence Maximization
We consider the problem of maximizing the spread of influence in a social
network by choosing a fixed number of initial seeds, formally referred to as
the influence maximization problem. It admits a -factor approximation
algorithm if the influence function is submodular. Otherwise, in the worst
case, the problem is NP-hard to approximate to within a factor of
. This paper studies whether this worst-case hardness result
can be circumvented by making assumptions about either the underlying network
topology or the cascade model. All of our assumptions are motivated by many
real life social network cascades.
First, we present strong inapproximability results for a very restricted
class of networks called the (stochastic) hierarchical blockmodel, a special
case of the well-studied (stochastic) blockmodel in which relationships between
blocks admit a tree structure. We also provide a dynamic-program based
polynomial time algorithm which optimally computes a directed variant of the
influence maximization problem on hierarchical blockmodel networks. Our
algorithm indicates that the inapproximability result is due to the
bidirectionality of influence between agent-blocks.
Second, we present strong inapproximability results for a class of influence
functions that are "almost" submodular, called 2-quasi-submodular. Our
inapproximability results hold even for any 2-quasi-submodular fixed in
advance. This result also indicates that the "threshold" between submodularity
and nonsubmodularity is sharp, regarding the approximability of influence
maximization.Comment: 53 pages, 20 figures; Conference short version - WINE 2017: The 13th
Conference on Web and Internet Economics; Journal full version - ACM:
Transactions on Computation Theory, 201
Think Globally, Act Locally: On the Optimal Seeding for Nonsubmodular Influence Maximization
We study the r-complex contagion influence maximization problem. In the influence maximization problem, one chooses a fixed number of initial seeds in a social network to maximize the spread of their influence. In the r-complex contagion model, each uninfected vertex in the network becomes infected if it has at least r infected neighbors.
In this paper, we focus on a random graph model named the stochastic hierarchical blockmodel, which is a special case of the well-studied stochastic blockmodel. When the graph is not exceptionally sparse, in particular, when each edge appears with probability omega (n^{-(1+1/r)}), under certain mild assumptions, we prove that the optimal seeding strategy is to put all the seeds in a single community. This matches the intuition that in a nonsubmodular cascade model placing seeds near each other creates synergy. However, it sharply contrasts with the intuition for submodular cascade models (e.g., the independent cascade model and the linear threshold model) in which nearby seeds tend to erode each others\u27 effects.
Finally, we show that this observation yields a polynomial time dynamic programming algorithm which outputs optimal seeds if each edge appears with a probability either in omega (n^{-(1+1/r)}) or in o (n^{-2})
Complexity, Algorithms, and Heuristics of Influence Maximization
People often adopt improved behaviors, products, or ideas through the influence of friends. This is modeled by emph{cascades}. One way to spread such positive elements through society is to identify those most influential agents---those that cause the maximum spread, and initiate the spread by seeding them. However, this strategy has a key difficulty: finding these influential seed nodes. This is difficult even if both the network structure and the way the cascade spreads are known. In emph{the influence maximization problem}, a central planner is given a graph and a limited budget , and he needs to pick seeds such that the expected total number of infected vertices in the graph at the end of the cascade is maximized. This problem plays a central role in viral marketing, outbreak detection, rumor controls, etc.
This thesis focuses on computational complexity, approximability and algorithm/heuristic design aspects of the influence maximization problem, with both emph{submodular} and emph{nonsubmodular} diffusion models. The first part of the thesis studies submodular influence maximization mainly in the computational complexity and algorithm analysis aspects, which includes some breakthroughs in understanding the approximability of submodular influence maximization and the theoretical performance of the well-studied greedy algorithm. The second part of the thesis focuses on nonsubmodular influence maximization. New sociologically founded nonsubmodular diffusion models are proposed, and we show how the seeding strategy for nonsubmodular diffusion models is fundamentally different compared to submodular diffusion models.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155221/1/bstao_1.pd
Incentive Ratios for Fairly Allocating Indivisible Goods: Simple Mechanisms Prevail
We study the problem of fairly allocating indivisible goods among strategic
agents. Amanatidis et al. show that truthfulness is incompatible with any
meaningful fairness notions. Thus we adopt the notion of incentive ratio, which
is defined as the ratio between the largest possible utility that an agent can
gain by manipulation and his utility in honest behavior under a given
mechanism. We select four of the most fundamental mechanisms in the literature
on discrete fair division, which are Round-Robin, a cut-and-choose mechanism of
Plaut and Roughgarden, Maximum-Nash-Welfare and Envy-Graph Procedure, and
obtain extensive results regarding the incentive ratios of them and their
variants.
For Round-Robin, we establish the incentive ratio of for additive and
subadditive cancelable valuations, the unbounded incentive ratio for cancelable
valuations, and the incentive ratios of and for
submodular and XOS valuations, respectively. Moreover, the incentive ratio is
unbounded for a variant that provides the -approximate maximum social
welfare guarantee. For the algorithm of Plaut and Roughgarden, the incentive
ratio is either unbounded or with lexicographic tie-breaking and is
with welfare maximizing tie-breaking. This separation exhibits the essential
role of tie-breaking rules in the design of mechanisms with low incentive
ratios. For Maximum-Nash-Welfare, the incentive ratio is unbounded.
Furthermore, the unboundedness can be bypassed by restricting agents to have a
strictly positive value for each good. For Envy-Graph Procedure, both of the
two possible ways of implementation lead to an unbounded incentive ratio.
Finally, we complement our results with a proof that the incentive ratio of
every mechanism satisfying envy-freeness up to one good is at least ,
and thus is larger than by a constant
Limitations of Greed: Influence Maximization in Undirected Networks Re-visited
We consider the influence maximization problem (selecting seeds in a
network maximizing the expected total influence) on undirected graphs under the
linear threshold model. On the one hand, we prove that the greedy algorithm
always achieves a -approximation, showing
that the greedy algorithm does slightly better on undirected graphs than the
generic bound which also applies to directed graphs. On the
other hand, we show that substantial improvement on this bound is impossible by
presenting an example where the greedy algorithm can obtain at most a approximation. This result stands in contrast to the
previous work on the independent cascade model. Like the linear threshold
model, the greedy algorithm obtains a -approximation on directed
graphs in the independent cascade model. However, Khanna and Lucier showed
that, in undirected graphs, the greedy algorithm performs substantially better:
a approximation for constant . Our results show
that, surprisingly, no such improvement occurs in the linear threshold model.
Finally, we show that, under the linear threshold model, the approximation
ratio is tight if 1) the graph is directed or 2) the
vertices are weighted. In other words, under either of these two settings, the
greedy algorithm cannot achieve a -approximation for
any positive function . The result in setting 2) is again in a sharp
contrast to Khanna and Lucier's -approximation result
for the independent cascade model, where the
approximation guarantee can be extended to the setting where vertices are
weighted. We also discuss extensions to more generalized settings including
those with edge-weighted graphs.Comment: 36 pages, 1 figure, accepted at AAMAS'20: International Conference on
Autonomous Agents and Multi-Agent System
The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation
We study the problem of allocating divisible resources among agents,
hopefully in a fair and efficient manner. With the presence of strategic
agents, additional incentive guarantees are also necessary, and the problem of
designing fair and efficient mechanisms becomes much less tractable. While the
maximum Nash welfare (MNW) mechanism has been proven to be prominent by
providing desirable fairness and efficiency guarantees as well as other
intuitive properties, no incentive property is known for it.
We show a surprising result that, when agents have piecewise constant value
density functions, the incentive ratio of the MNW mechanism is for cake
cutting, where the incentive ratio of a mechanism is defined as the ratio
between the largest possible utility that an agent can gain by manipulation and
his utility in honest behavior. Remarkably, this result holds even without the
free disposal assumption, which is hard to get rid of in the design of truthful
cake cutting mechanisms. We also show that the MNW mechanism is group
strategyproof when agents have piecewise uniform value density functions.
Moreover, we show that, for cake cutting, the Partial Allocation (PA)
mechanism proposed by Cole et al., which is truthful and -MNW for
homogeneous divisible items, has an incentive ratio between
and when randomization is allowed, can be turned to be truthful in expectation.
Given two alternatives for a trade-off between incentive ratio and Nash welfare
provided by the MNW and PA mechanisms, we establish an interpolation between
them for both cake cutting and homogeneous divisible items.
Finally, we study the existence of fair mechanisms with a low incentive ratio
in the connected pieces setting. We show that any envy-free cake cutting
mechanism with the connected pieces constraint has an incentive ratio of at
least
Adaptive Greedy versus Non-adaptive Greedy for Influence Maximization
We consider the \emph{adaptive influence maximization problem}: given a
network and a budget , iteratively select seeds in the network to
maximize the expected number of adopters. In the \emph{full-adoption feedback
model}, after selecting each seed, the seed-picker observes all the resulting
adoptions. In the \emph{myopic feedback model}, the seed-picker only observes
whether each neighbor of the chosen seed adopts. Motivated by the extreme
success of greedy-based algorithms/heuristics for influence maximization, we
propose the concept of \emph{greedy adaptivity gap}, which compares the
performance of the adaptive greedy algorithm to its non-adaptive counterpart.
Our first result shows that, for submodular influence maximization, the
adaptive greedy algorithm can perform up to a -fraction worse than the
non-adaptive greedy algorithm, and that this ratio is tight. More specifically,
on one side we provide examples where the performance of the adaptive greedy
algorithm is only a fraction of the performance of the non-adaptive
greedy algorithm in four settings: for both feedback models and both the
\emph{independent cascade model} and the \emph{linear threshold model}. On the
other side, we prove that in any submodular cascade, the adaptive greedy
algorithm always outputs a -approximation to the expected number of
adoptions in the optimal non-adaptive seed choice. Our second result shows
that, for the general submodular cascade model with full-adoption feedback, the
adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by
an unbounded factor. Finally, we propose a risk-free variant of the adaptive
greedy algorithm that always performs no worse than the non-adaptive greedy
algorithm.Comment: 26 pages, 0 figure, accepted at AAAI'20: Thirty-Fourth AAAI
Conference on Artificial Intelligenc
Outsourcing Computation: the Minimal Refereed Mechanism
We consider a setting where a verifier with limited computation power
delegates a resource intensive computation task---which requires a
computation tableau---to two provers where the provers are rational in that
each prover maximizes their own payoff---taking into account losses incurred by
the cost of computation. We design a mechanism called the Minimal Refereed
Mechanism (MRM) such that if the verifier has time and
space computation power, then both provers will provide a
honest result without the verifier putting any effort to verify the results.
The amount of computation required for the provers (and thus the cost) is a
multiplicative -factor more than the computation itself, making this
schema efficient especially for low-space computations.Comment: 17 pages, 1 figure; WINE 2019: The 15th Conference on Web and
Internet Economic
On computational complexity of plane curve invariants
The theory of generic smooth closed plane curves initiated by Vladimir Arnold is a beautiful fusion of topology, combinatorics, and analysis. The theory remains fairly undeveloped. We review existing methods to describe generic smooth closed plane curves combinatorially, introduce a new one, and give an algorithm for efficient computation of Arnold's invariants. Our results provide a good source of future research projects that involve computer experiments with plane curves. The reader is not required to have background in topology and even undergraduate students with basic knowledge of differential geometry and graph theory will easily understand our paper.Published versio
Adaptive Greedy versus Non-Adaptive Greedy for Influence Maximization
We consider the adaptive influence maximization problem: given a network and a budget k, iteratively select k seeds in the network to maximize the expected number of adopters. In the full-adoption feedback model, after selecting each seed, the seed-picker observes all the resulting adoptions. In the myopic feedback model, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of greedy adaptivity gap, which compares the performance of the adaptive greedy algorithm to its non-adaptive counterpart. Our first result shows that, for submodular influence maximization, the adaptive greedy algorithm can perform up to a (1-1/e)-fraction worse than the non-adaptive greedy algorithm, and that this ratio is tight. More specifically, on one side we provide examples where the performance of the adaptive greedy algorithm is only a (1-1/e) fraction of the performance of the non-adaptive greedy algorithm in four settings: for both feedback models and both the independent cascade model and the linear threshold model. On the other side, we prove that in any submodular cascade, the adaptive greedy algorithm always outputs a (1-1/e)-approximation to the expected number of adoptions in the optimal non-adaptive seed choice. Our second result shows that, for the general submodular cascade model with full-adoption feedback, the adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by an unbounded factor. Finally, we propose a risk-free variant of the adaptive greedy algorithm that always performs no worse than the non-adaptive greedy algorithm