222 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On linear, fractional, and submodular optimization
In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree
The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences
We study the problem of fairly allocating indivisible goods (positively
valued items) and chores (negatively valued items) among agents with decreasing
marginal utilities over items. Our focus is on instances where all the agents
have simple preferences; specifically, we assume the marginal value of an item
can be either , or some positive integer . Under this assumption, we
present an efficient algorithm to compute leximin allocations for a broad class
of valuation functions we call order-neutral submodular valuations.
Order-neutral submodular valuations strictly contain the well-studied class of
additive valuations but are a strict subset of the class of submodular
valuations. We show that these leximin allocations are Lorenz dominating and
approximately proportional. We also show that, under further restriction to
additive valuations, these leximin allocations are approximately envy-free and
guarantee each agent their maxmin share. We complement this algorithmic result
with a lower bound showing that the problem of computing leximin allocations is
NP-hard when is a rational number
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results
Fair resource allocation is an important problem in many real-world
scenarios, where resources such as goods and chores must be allocated among
agents. In this survey, we delve into the intricacies of fair allocation,
focusing specifically on the challenges associated with indivisible resources.
We define fairness and efficiency within this context and thoroughly survey
existential results, algorithms, and approximations that satisfy various
fairness criteria, including envyfreeness, proportionality, MMS, and their
relaxations. Additionally, we discuss algorithms that achieve fairness and
efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study
the computational complexity of these algorithms, the likelihood of finding
fair allocations, and the price of fairness for each fairness notion. We also
cover mixed instances of indivisible and divisible items and investigate
different valuation and allocation settings. By summarizing the
state-of-the-art research, this survey provides valuable insights into fair
resource allocation of indivisible goods and chores, highlighting computational
complexities, fairness guarantees, and trade-offs between fairness and
efficiency. It serves as a foundation for future advancements in this vital
field
A Tight Competitive Ratio for Online Submodular Welfare Maximization
In this paper we consider the online Submodular Welfare (SW) problem. In this
problem we are given bidders each equipped with a general (not necessarily
monotone) submodular utility and items that arrive online. The goal is to
assign each item, once it arrives, to a bidder or discard it, while maximizing
the sum of utilities. When an adversary determines the items' arrival order we
present a simple randomized algorithm that achieves a tight competitive ratio
of \nicefrac{1}{4}. The algorithm is a specialization of an algorithm due to
[Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best
known competitive ratio of to the problem. When
the items' arrival order is uniformly random, we present a competitive ratio of
, improving the previously known \nicefrac{1}{4} guarantee.
Our approach for the latter result is based on a better analysis of the
(offline) Residual Random Greedy (RRG) algorithm of
[Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of
independent interest
A Tight Competitive Ratio for Online Submodular Welfare Maximization
In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given n bidders each equipped with a general non-negative (not necessarily monotone) submodular utility and m items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items\u27 arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of 1/4. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of 3-2?2? 0.171573 to the problem. When the items\u27 arrival order is uniformly random, we present a competitive ratio of ? 0.27493, improving the previously known 1/4 guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
A constant factor approximation for Nash social welfare with subadditive valuations
We present a constant-factor approximation algorithm for the Nash social
welfare maximization problem with subadditive valuations accessible via demand
queries. More generally, we propose a template for NSW optimization by solving
a configuration-type LP and using a rounding procedure for (utilitarian) social
welfare as a blackbox, which could be applicable to other variants of the
problem
Social Media Influencers- A Review of Operations Management Literature
This literature review provides a comprehensive survey of research on Social Media
Influencers (SMIs) across the fields of SMIs in marketing, seeding strategies, influence
maximization and applications of SMIs in society. Specifically, we focus on examining the
methods employed by researchers to reach their conclusions. Through our analysis, we
identify opportunities for future research that align with emerging areas and unexplored
territories related to theory, context, and methodology. This approach offers a fresh
perspective on existing research, paving the way for more effective and impactful studies in
the future. Additionally, gaining a deeper understanding of the underlying principles and
methodologies of these concepts enables more informed decision-making when
implementing these strategie
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