8 research outputs found
On the Economic Efficiency of the Combinatorial Clock Auction
Since the 1990s spectrum auctions have been implemented world-wide. This has
provided for a practical examination of an assortment of auction mechanisms
and, amongst these, two simultaneous ascending price auctions have proved to be
extremely successful. These are the simultaneous multiround ascending auction
(SMRA) and the combinatorial clock auction (CCA). It has long been known that,
for certain classes of valuation functions, the SMRA provides good theoretical
guarantees on social welfare. However, no such guarantees were known for the
CCA.
In this paper, we show that CCA does provide strong guarantees on social
welfare provided the price increment and stopping rule are well-chosen. This is
very surprising in that the choice of price increment has been used primarily
to adjust auction duration and the stopping rule has attracted little
attention. The main result is a polylogarithmic approximation guarantee for
social welfare when the maximum number of items demanded by a
bidder is fixed. Specifically, we show that either the revenue of the CCA is at
least an -fraction of
the optimal welfare or the welfare of the CCA is at least an
-fraction of the optimal welfare, where
is the number of bidders and is the number of items. As a corollary, the
welfare ratio -- the worst case ratio between the social welfare of the optimum
allocation and the social welfare of the CCA allocation -- is at most
. We emphasize that this latter
result requires no assumption on bidders valuation functions. Finally, we prove
that such a dependence on is necessary. In particular, we show
that the welfare ratio of the CCA is at least
An Algorithmic Theory of Integer Programming
We study the general integer programming problem where the number of
variables is a variable part of the input. We consider two natural
parameters of the constraint matrix : its numeric measure and its
sparsity measure . We show that integer programming can be solved in time
, where is some computable function of the
parameters and , and is the binary encoding length of the input. In
particular, integer programming is fixed-parameter tractable parameterized by
and , and is solvable in polynomial time for every fixed and .
Our results also extend to nonlinear separable convex objective functions.
Moreover, for linear objectives, we derive a strongly-polynomial algorithm,
that is, with running time , independent of the rest of
the input data.
We obtain these results by developing an algorithmic framework based on the
idea of iterative augmentation: starting from an initial feasible solution, we
show how to quickly find augmenting steps which rapidly converge to an optimum.
A central notion in this framework is the Graver basis of the matrix , which
constitutes a set of fundamental augmenting steps. The iterative augmentation
idea is then enhanced via the use of other techniques such as new and improved
bounds on the Graver basis, rapid solution of integer programs with bounded
variables, proximity theorems and a new proximity-scaling algorithm, the notion
of a reduced objective function, and others.
As a consequence of our work, we advance the state of the art of solving
block-structured integer programs. In particular, we develop near-linear time
algorithms for -fold, tree-fold, and -stage stochastic integer programs.
We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified
proximity-scaling algorith
Tight Lower Bounds for Block-Structured Integer Programs
We study fundamental block-structured integer programs called tree-fold and
multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent
blocks linked together by few constraints in a recursive pattern; and
transposing their constraint matrix yields multi-stage IPs. The
state-of-the-art algorithms to solve these IPs have an exponential gap in their
running times, making it natural to ask whether this gap is inherent. We answer
this question affirmative. Assuming the Exponential Time Hypothesis, we prove
lower bounds showing that the exponential difference is necessary, and that the
known algorithms are near optimal. Moreover, we prove unconditional lower
bounds on the norms of the Graver basis, a fundamental building block of all
known algorithms to solve these IPs. This shows that none of the current
approaches can be improved beyond this bound
New Results in Integer and Lattice Programming
An integer program (IP) is a problem of the form , where , , , and is a separable convex objective function.
The problem of finding an optimal solution for an integer program is known as integer programming.
Integer programming is NP-hard in general, though several algorithms exist:
Lenstra provided an algorithm that is polynomial if the dimension is fixed.
For variable dimension, the best known algorithm depends linearly on , and exponentially on the number of equalities as well as the largest absolute value of an entry in the matrix .
The first part of this thesis considers integer programming for variable dimensions and sparse matrices.
We measure the sparsity of a matrix by the tree-depth of the dual graph of .
A typical example for these integer programs are -fold IPs, used for scheduling and social choice problems.
We obtain the currently fastest fixed-parameter tractable algorithm with parameters tree-depth and the largest absolute value of the entries in .
The running time we achieve is near-linear in the dimension.
With a slightly worse running time, we are able to show that -fold integer programs of constant block size can be solved in strongly polynomial time.
Assuming the exponential time hypothesis, we complement these results with a lower bound on the parameter dependency that almost matches the parameter dependency of the running time.
As a consequence, we provide the currently strongest lower bound for -fold integer programs.
Another problem closely related to integer programming is the closest vector problem.
A lattice is a discrete additive subgroup of .
The closest vector problem (CVP) asks for a lattice point closest to a given target vector.
An important tool for solving the closest vector problem is the Voronoi cell \vc of a lattice , which is the set of all points for which is a closest lattice point.
It is a polytope whose facets are induced by a set of lattice vectors, the Voronoi relevant vectors.
A generic lattice has exponentially many Voronoi relevant vectors, leading to exponential space for certain CVP algorithms.
In the second part of this thesis, we introduce the notion of a -compact lattice basis that facilitates to represent the Voronoi relevant vectors with coefficients bounded by .
Such a basis allows to reduce the space requirement of Micciancio's \& Voulgaris' algorithm for the closest vector problem from exponential to polynomial, while the running time becomes exponential in .
We show that for every lattice an -compact basis exists, but there are lattices for which we cannot choose . If the Voronoi cell is a zonotope, we can choose , providing a single-exponential time and polynomial space algorithm for CVP, assuming a -compact basis is known.
Deciding whether a given lattice has a certain structure that helps to solve the closest vector problem more efficiently is a reappearing and non-trivial problem.
The third part of this thesis is concerned with the specific structure of having an orthonormal basis.
We show that this problem belongs to NP co-NP.
Moreover, it can be reduced to solving a single closest vector problem.
We also show that if a separation oracle for the Voronoi cell is provided, CVP is solvable in polynomial time
On compact representations of Voronoi cells of lattices
In a seminal work, Micciancio & Voulgaris (2010) described a deterministic single-exponential time algorithm for the Closest Vector Problem (CVP) on lattices. It is based on the computation of the Voronoi cell of the given lattice and thus may need exponential space as well. We address the major open question whether there exists such an algorithm that requires only polynomial space. To this end, we define a lattice basis to be c-compact if every facet normal of the Voronoi cell is a linear combination of the basis vectors using coefficients that are bounded by c in absolute value. Given such a basis, we get a polynomial space algorithm for CVP whose running time naturally depends on c. Thus, our main focus is the behavior of the smallest possible value of c, with the following results: There always exist c-compact bases, where c is bounded by n^2 for an n-dimension lattice; there are lattices not admitting a c-compact basis with c growing sublinearly with the dimension; and every lattice with a zonotopal Voronoi cell has a 1-compact basis