1,683 research outputs found
Maximizing Symmetric Submodular Functions
Symmetric submodular functions are an important family of submodular
functions capturing many interesting cases including cut functions of graphs
and hypergraphs. Maximization of such functions subject to various constraints
receives little attention by current research, unlike similar minimization
problems which have been widely studied. In this work, we identify a few
submodular maximization problems for which one can get a better approximation
for symmetric objectives than the state of the art approximation for general
submodular functions.
We first consider the problem of maximizing a non-negative symmetric
submodular function subject to a
down-monotone solvable polytope . For
this problem we describe an algorithm producing a fractional solution of value
at least , where is the optimal integral solution.
Our second result considers the problem for a
non-negative symmetric submodular function . For this problem, we give an approximation ratio that depends on
the value and is always at least . Our method can
also be applied to non-negative non-symmetric submodular functions, in which
case it produces approximation, improving over the best known
result for this problem. For unconstrained maximization of a non-negative
symmetric submodular function we describe a deterministic linear-time
-approximation algorithm. Finally, we give a -approximation algorithm for Submodular Welfare with players having
identical non-negative submodular utility functions, and show that this is the
best possible approximation ratio for the problem.Comment: 31 pages, an extended abstract appeared in ESA 201
On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives
We study a class of procurement auctions with a budget constraint, where an
auctioneer is interested in buying resources or services from a set of agents.
Ideally, the auctioneer would like to select a subset of the resources so as to
maximize his valuation function, without exceeding a given budget. As the
resources are owned by strategic agents however, our overall goal is to design
mechanisms that are truthful, budget-feasible, and obtain a good approximation
to the optimal value. Budget-feasibility creates additional challenges, making
several approaches inapplicable in this setting. Previous results on
budget-feasible mechanisms have considered mostly monotone valuation functions.
In this work, we mainly focus on symmetric submodular valuations, a prominent
class of non-monotone submodular functions that includes cut functions. We
begin first with a purely algorithmic result, obtaining a
-approximation for maximizing symmetric submodular functions
under a budget constraint. We view this as a standalone result of independent
interest, as it is the best known factor achieved by a deterministic algorithm.
We then proceed to propose truthful, budget feasible mechanisms (both
deterministic and randomized), paying particular attention on the Budgeted Max
Cut problem. Our results significantly improve the known approximation ratios
for these objectives, while establishing polynomial running time for cases
where only exponential mechanisms were known. At the heart of our approach lies
an appropriate combination of local search algorithms with results for monotone
submodular valuations, applied to the derived local optima.Comment: A conference version appears in WINE 201
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