9 research outputs found
Constrained Monotone Function Maximization and the Supermodular Degree
The problem of maximizing a constrained monotone set function has many
practical applications and generalizes many combinatorial problems.
Unfortunately, it is generally not possible to maximize a monotone set function
up to an acceptable approximation ratio, even subject to simple constraints.
One highly studied approach to cope with this hardness is to restrict the set
function. An outstanding disadvantage of imposing such a restriction on the set
function is that no result is implied for set functions deviating from the
restriction, even slightly. A more flexible approach, studied by Feige and
Izsak, is to design an approximation algorithm whose approximation ratio
depends on the complexity of the instance, as measured by some complexity
measure. Specifically, they introduced a complexity measure called supermodular
degree, measuring deviation from submodularity, and designed an algorithm for
the welfare maximization problem with an approximation ratio that depends on
this measure.
In this work, we give the first (to the best of our knowledge) algorithm for
maximizing an arbitrary monotone set function, subject to a k-extendible
system. This class of constraints captures, for example, the intersection of
k-matroids (note that a single matroid constraint is sufficient to capture the
welfare maximization problem). Our approximation ratio deteriorates gracefully
with the complexity of the set function and k. Our work can be seen as
generalizing both the classic result of Fisher, Nemhauser and Wolsey, for
maximizing a submodular set function subject to a k-extendible system, and the
result of Feige and Izsak for the welfare maximization problem. Moreover, when
our algorithm is applied to each one of these simpler cases, it obtains the
same approximation ratio as of the respective original work.Comment: 23 page
Cooperative Games with Bounded Dependency Degree
Cooperative games provide a framework to study cooperation among
self-interested agents. They offer a number of solution concepts describing how
the outcome of the cooperation should be shared among the players.
Unfortunately, computational problems associated with many of these solution
concepts tend to be intractable---NP-hard or worse. In this paper, we
incorporate complexity measures recently proposed by Feige and Izsak (2013),
called dependency degree and supermodular degree, into the complexity analysis
of cooperative games. We show that many computational problems for cooperative
games become tractable for games whose dependency degree or supermodular degree
are bounded. In particular, we prove that simple games admit efficient
algorithms for various solution concepts when the supermodular degree is small;
further, we show that computing the Shapley value is always in FPT with respect
to the dependency degree. Finally, we note that, while determining the
dependency among players is computationally hard, there are efficient
algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape
A Unifying Hierarchy of Valuations with Complements and Substitutes
We introduce a new hierarchy over monotone set functions, that we refer to as
(Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, - (where is the total number of
items) captures all monotone functions. The lowest level, -,
captures all monotone submodular functions, and more generally, the class of
functions known as . Every monotone function that has a positive
hypergraph representation of rank (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in -. Every
monotone function that has supermodular degree (in the sense defined by
Feige and Izsak [ITCS 2013]) is in -. In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
-.
One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of if all
players hold valuation functions in -. The other is an upper
bound of on the price of anarchy of simultaneous first price auctions.
Being in - can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
(Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research
Welfare Maximization and the Supermodular Degree
Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items, the welfare maximization problem requires that every item be allocated to exactly one player, and one wishes to maximize the sum of values obtained by the players, as computed by applying the respective valuation function to the bundle of items allocated to the player. This problem in its full generality is NP-hard, and moreover, at least as hard to approximate as set packing. Better approximation guarantees are known for restricted classes of valuation functions. In this work we introduce a new parameter, the supermodular degree of a valuation function, which is a measure for the extent to which the function exhibits supermodular behavior. We design an approximation algorithm for the welfare maximization problem whose approximation guarantee is linear in the supermodular degree of the underlying valuation functions
Committee Selection with Intraclass and Interclass Synergies
Voting is almost never done in void, as usually there are some relations between the alternatives on which the voters vote on. These relations shall be taken into consideration when selecting a winning committee of some given multiwinner election. As taking into account all possible relations between the alternatives is generally computationally intractable, in this paper we consider classes of alternatives; intuitively, the number of classes is significantly smaller than the number of alternatives, and thus there is some hope in reaching computational tractability. We model both intraclass relations and interclass relations by functions, which we refer to as synergy functions, and study the computational complexity of identifying the best committee, taking into account those synergy functions. Our model accommodates both positive and negative relations between alternatives; further, our efficient algorithms can also deal with a rich class of diversity wishes, which we show how to model using synergy functions