22 research outputs found
Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design
We study a type of reverse (procurement) auction problems in the presence of
budget constraints. The general algorithmic problem is to purchase a set of
resources, which come at a cost, so as not to exceed a given budget and at the
same time maximize a given valuation function. This framework captures the
budgeted version of several well known optimization problems, and when the
resources are owned by strategic agents the goal is to design truthful and
budget feasible mechanisms, i.e. elicit the true cost of the resources and
ensure the payments of the mechanism do not exceed the budget. Budget
feasibility introduces more challenges in mechanism design, and we study
instantiations of this problem for certain classes of submodular and XOS
valuation functions. We first obtain mechanisms with an improved approximation
ratio for weighted coverage valuations, a special class of submodular functions
that has already attracted attention in previous works. We then provide a
general scheme for designing randomized and deterministic polynomial time
mechanisms for a class of XOS problems. This class contains problems whose
feasible set forms an independence system (a more general structure than
matroids), and some representative problems include, among others, finding
maximum weighted matchings, maximum weighted matroid members, and maximum
weighted 3D-matchings. For most of these problems, only randomized mechanisms
with very high approximation ratios were known prior to our results
On Finding Maximum Cardinality Subset of Vectors with a Constraint on Normalized Squared Length of Vectors Sum
In this paper, we consider the problem of finding a maximum cardinality
subset of vectors, given a constraint on the normalized squared length of
vectors sum. This problem is closely related to Problem 1 from (Eremeev,
Kel'manov, Pyatkin, 2016). The main difference consists in swapping the
constraint with the optimization criterion.
We prove that the problem is NP-hard even in terms of finding a feasible
solution. An exact algorithm for solving this problem is proposed. The
algorithm has a pseudo-polynomial time complexity in the special case of the
problem, where the dimension of the space is bounded from above by a constant
and the input data are integer. A computational experiment is carried out,
where the proposed algorithm is compared to COINBONMIN solver, applied to a
mixed integer quadratic programming formulation of the problem. The results of
the experiment indicate superiority of the proposed algorithm when the
dimension of Euclidean space is low, while the COINBONMIN has an advantage for
larger dimensions.Comment: To appear in Proceedings of the 6th International Conference on
Analysis of Images, Social Networks, and Texts (AIST'2017
A 0.821-ratio purely combinatorial algorithm for maximum k-vertex cover in bipartite graphs
We study the polynomial time approximation of the max k-vertex cover problem in bipartite graphs and propose a purely combinatorial algorithm that beats the only such known algorithm, namely the greedy approach. We present a computer-assisted analysis of our algorithm, establishing that the worst case approximation guarantee is bounded below by 0.821. © Springer-Verlag Berlin Heidelberg 2016
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
Stable marriage and roommates problems with restricted edges: complexity and approximability
In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.
Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.
Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs
On the Hardness of Energy Minimisation for Crystal Structure Prediction
Crystal Structure Prediction (csp) is one of the central and most challenging
problems in materials science and computational chemistry. In csp, the goal is
to find a configuration of ions in 3D space that yields the lowest potential
energy. Finding an efficient procedure to solve this complex optimisation
question is a well known open problem. Due to the exponentially large search
space, the problem has been referred in several materials-science papers as
"NP-Hard and very challenging" without a formal proof. This paper fills a gap
in the literature providing the first set of formally proven NP-Hardness
results for a variant of csp with various realistic constraints. In particular,
we focus on the problem of removal: the goal is to find a substructure with
minimal potential energy, by removing a subset of the ions. Our main
contributions are NP-Hardness results for the csp removal problem, new
embeddings of combinatorial graph problems into geometrical settings, and a
more systematic exploration of the energy function to reveal the complexity of
csp. In a wider context, our results contribute to the analysis of
computational problems for weighted graphs embedded into the three-dimensional
Euclidean space.Comment: Short version published in SOFSEM 2020, full version to be published
in Fundamenta Informatica