261 research outputs found
Sponsored Search, Market Equilibria, and the Hungarian Method
Matching markets play a prominent role in economic theory. A prime example of
such a market is the sponsored search market. Here, as in other markets of that
kind, market equilibria correspond to feasible, envy free, and bidder optimal
outcomes. For settings without budgets such an outcome always exists and can be
computed in polynomial-time by the so-called Hungarian Method. Moreover, every
mechanism that computes such an outcome is incentive compatible. We show that
the Hungarian Method can be modified so that it finds a feasible, envy free,
and bidder optimal outcome for settings with budgets. We also show that in
settings with budgets no mechanism that computes such an outcome can be
incentive compatible for all inputs. For inputs in general position, however,
the presented mechanism---as any other mechanism that computes such an outcome
for settings with budgets---is incentive compatible
Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time
In the decremental single-source shortest paths (SSSP) problem we want to
maintain the distances between a given source node and every other node in
an -node -edge graph undergoing edge deletions. While its static
counterpart can be solved in near-linear time, this decremental problem is much
more challenging even in the undirected unweighted case. In this case, the
classic total update time of Even and Shiloach [JACM 1981] has been the
fastest known algorithm for three decades. At the cost of a
-approximation factor, the running time was recently improved to
by Bernstein and Roditty [SODA 2011]. In this paper, we bring the
running time down to near-linear: We give a -approximation
algorithm with expected total update time, thus obtaining
near-linear time. Moreover, we obtain time for the weighted
case, where the edge weights are integers from to . The only prior work
on weighted graphs in time is the -time algorithm by
Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with
quasi-polynomial edge weights. The expected running time bound of our algorithm
holds against an oblivious adversary.
In contrast to the previous results which rely on maintaining a sparse
emulator, our algorithm relies on maintaining a so-called sparse -hop set introduced by Cohen [JACM 2000] in the PRAM literature. An
-hop set of a graph is a set of weighted edges
such that the distance between any pair of nodes in can be
-approximated by their -hop distance (given by a path
containing at most edges) on . Our algorithm can maintain
an -hop set of near-linear size in near-linear time under
edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper
was presented at the 55th IEEE Symposium on Foundations of Computer Science
(FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character
Valuation Compressions in VCG-Based Combinatorial Auctions
The focus of classic mechanism design has been on truthful direct-revelation
mechanisms. In the context of combinatorial auctions the truthful
direct-revelation mechanism that maximizes social welfare is the VCG mechanism.
For many valuation spaces computing the allocation and payments of the VCG
mechanism, however, is a computationally hard problem. We thus study the
performance of the VCG mechanism when bidders are forced to choose bids from a
subspace of the valuation space for which the VCG outcome can be computed
efficiently. We prove improved upper bounds on the welfare loss for
restrictions to additive bids and upper and lower bounds for restrictions to
non-additive bids. These bounds show that the welfare loss increases in
expressiveness. All our bounds apply to equilibrium concepts that can be
computed in polynomial time as well as to learning outcomes
Fully Dynamic Single-Source Reachability in Practice: An Experimental Study
Given a directed graph and a source vertex, the fully dynamic single-source
reachability problem is to maintain the set of vertices that are reachable from
the given vertex, subject to edge deletions and insertions. It is one of the
most fundamental problems on graphs and appears directly or indirectly in many
and varied applications. While there has been theoretical work on this problem,
showing both linear conditional lower bounds for the fully dynamic problem and
insertions-only and deletions-only upper bounds beating these conditional lower
bounds, there has been no experimental study that compares the performance of
fully dynamic reachability algorithms in practice. Previous experimental
studies in this area concentrated only on the more general all-pairs
reachability or transitive closure problem and did not use real-world dynamic
graphs.
In this paper, we bridge this gap by empirically studying an extensive set of
algorithms for the single-source reachability problem in the fully dynamic
setting. In particular, we design several fully dynamic variants of well-known
approaches to obtain and maintain reachability information with respect to a
distinguished source. Moreover, we extend the existing insertions-only or
deletions-only upper bounds into fully dynamic algorithms. Even though the
worst-case time per operation of all the fully dynamic algorithms we evaluate
is at least linear in the number of edges in the graph (as is to be expected
given the conditional lower bounds) we show in our extensive experimental
evaluation that their performance differs greatly, both on generated as well as
on real-world instances
- …