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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Fair Knapsack
We study the following multiagent variant of the knapsack problem. We are
given a set of items, a set of voters, and a value of the budget; each item is
endowed with a cost and each voter assigns to each item a certain value. The
goal is to select a subset of items with the total cost not exceeding the
budget, in a way that is consistent with the voters' preferences. Since the
preferences of the voters over the items can vary significantly, we need a way
of aggregating these preferences, in order to select the socially best valid
knapsack. We study three approaches to aggregating voters' preferences, which
are motivated by the literature on multiwinner elections and fair allocation.
This way we introduce the concepts of individually best, diverse, and fair
knapsack. We study the computational complexity (including parameterized
complexity, and complexity under restricted domains) of the aforementioned
multiagent variants of knapsack.Comment: Extended abstract will appear in Proc. of 33rd AAAI 201
Sparse grid quadrature on products of spheres
We examine sparse grid quadrature on weighted tensor products (WTP) of
reproducing kernel Hilbert spaces on products of the unit sphere, in the case
of worst case quadrature error for rules with arbitrary quadrature weights. We
describe a dimension adaptive quadrature algorithm based on an algorithm of
Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowski's
WTP algorithm (1999), here called the WW algorithm. We prove that the dimension
adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no
greater in cost than the WW algorithm. Both algorithms therefore have the
optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and
Wozniakowski (1999). A numerical example shows that, even though the asymptotic
convergence rate is optimal, if the dimension weights decay slowly enough, and
the dimensionality of the problem is large enough, the initial convergence of
the dimension adaptive algorithm can be slow.Comment: 34 pages, 6 figures. Accepted 7 January 2015 for publication in
Numerical Algorithms. Revised at page proof stage to (1) update email
address; (2) correct the accent on "Wozniakowski" on p. 7; (3) update
reference 2; (4) correct references 3, 18 and 2
Truthful Assignment without Money
We study the design of truthful mechanisms that do not use payments for the
generalized assignment problem (GAP) and its variants. An instance of the GAP
consists of a bipartite graph with jobs on one side and machines on the other.
Machines have capacities and edges have values and sizes; the goal is to
construct a welfare maximizing feasible assignment. In our model of private
valuations, motivated by impossibility results, the value and sizes on all
job-machine pairs are public information; however, whether an edge exists or
not in the bipartite graph is a job's private information.
We study several variants of the GAP starting with matching. For the
unweighted version, we give an optimal strategyproof mechanism; for maximum
weight bipartite matching, however, we show give a 2-approximate strategyproof
mechanism and show by a matching lowerbound that this is optimal. Next we study
knapsack-like problems, which are APX-hard. For these problems, we develop a
general LP-based technique that extends the ideas of Lavi and Swamy to reduce
designing a truthful mechanism without money to designing such a mechanism for
the fractional version of the problem, at a loss of a factor equal to the
integrality gap in the approximation ratio. We use this technique to obtain
strategyproof mechanisms with constant approximation ratios for these problems.
We then design an O(log n)-approximate strategyproof mechanism for the GAP by
reducing, with logarithmic loss in the approximation, to our solution for the
value-invariant GAP. Our technique may be of independent interest for designing
truthful mechanisms without money for other LP-based problems.Comment: Extended abstract appears in the 11th ACM Conference on Electronic
Commerce (EC), 201
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
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