1,957 research outputs found
Online Dependent Rounding Schemes
We study the abstract problem of rounding fractional bipartite -matchings
online. The input to the problem is an unknown fractional bipartite
-matching, exposed node-by-node on one side. The objective is to maximize
the \emph{rounding ratio} of the output matching , which is the
minimum over all fractional -matchings , and edges , of the
ratio . In offline settings, many dependent rounding
schemes achieving a ratio of one and strong negative correlation properties are
known (e.g., Gandhi et al., J.ACM'06 and Chekuri et al., FOCS'10), and have
found numerous applications. Motivated by online applications, we present
\emph{online dependent-rounding schemes} (ODRSes) for -matching.
For the special case of uniform matroids (single offline node), we present a
simple online algorithm with a rounding ratio of one. Interestingly, we show
that our algorithm yields \emph{the same distribution} as its classic offline
counterpart, pivotal sampling (Srinivasan, FOCS'01), and so inherits the
latter's strong correlation properties. In arbitrary bipartite graphs, an
online rounding ratio of one is impossible, and we show that a combination of
our uniform matroid ODRS with repeated invocations of \emph{offline} contention
resolution schemes (CRSes) yields a rounding ratio of . Our
main technical contribution is an ODRS breaking this pervasive bound, yielding
rounding ratios of and for -matchings and simple matchings,
respectively. We obtain these results by grouping nodes and using CRSes for
negatively-correlated distributions, together with a new method we call
\emph{group discount and individual markup}, analyzed using the theory of
negative association. We present a number of applications of our ODRSes to
online edge coloring, several stochastic optimization problems, and algorithmic
fairness
Dependent rounding with strong negative-correlation, and scheduling on unrelated machines to minimize completion time
We describe a new dependent-rounding algorithmic framework for bipartite
graphs. Given a fractional assignment of values to edges of graph , the algorithms return an integral solution such that each
right-node has at most one neighboring edge with , and
where the variables also satisfy broad nonpositive-correlation
properties. In particular, for any edges sharing a left-node , the variables have strong negative-correlation
properties, i.e. the expectation of is significantly below
.
This algorithm is a refinement of a dependent-rounding algorithm of Im \&
Shadloo (2020) based on simulation of Poisson processes. Our algorithm allows
greater flexibility, in particular, it allows ``irregular'' fractional
assignments, and it gives more refined bounds on the negative correlation.
Dependent rounding schemes with negative correlation properties have been
used for approximation algorithms for job-scheduling on unrelated machines to
minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im &
Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm,
among other improvements, we obtain a -approximation for this problem.
This significantly improves over the prior -approximation ratio of Im &
Li (2023)
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
Improved Approximation Algorithms for Stochastic Matching
In this paper we consider the Stochastic Matching problem, which is motivated
by applications in kidney exchange and online dating. We are given an
undirected graph in which every edge is assigned a probability of existence and
a positive profit, and each node is assigned a positive integer called timeout.
We know whether an edge exists or not only after probing it. On this random
graph we are executing a process, which one-by-one probes the edges and
gradually constructs a matching. The process is constrained in two ways: once
an edge is taken it cannot be removed from the matching, and the timeout of
node upper-bounds the number of edges incident to that can be probed.
The goal is to maximize the expected profit of the constructed matching.
For this problem Bansal et al. (Algorithmica 2012) provided a
-approximation algorithm for bipartite graphs, and a -approximation for
general graphs. In this work we improve the approximation factors to
and , respectively.
We also consider an online version of the bipartite case, where one side of
the partition arrives node by node, and each time a node arrives we have to
decide which edges incident to we want to probe, and in which order. Here
we present a -approximation, improving on the -approximation of
Bansal et al.
The main technical ingredient in our result is a novel way of probing edges
according to a random but non-uniform permutation. Patching this method with an
algorithm that works best for large probability edges (plus some additional
ideas) leads to our improved approximation factors
Energy Efficient Scheduling via Partial Shutdown
Motivated by issues of saving energy in data centers we define a collection
of new problems referred to as "machine activation" problems. The central
framework we introduce considers a collection of machines (unrelated or
related) with each machine having an {\em activation cost} of . There
is also a collection of jobs that need to be performed, and is
the processing time of job on machine . We assume that there is an
activation cost budget of -- we would like to {\em select} a subset of
the machines to activate with total cost and {\em find} a schedule
for the jobs on the machines in minimizing the makespan (or any other
metric).
For the general unrelated machine activation problem, our main results are
that if there is a schedule with makespan and activation cost then we
can obtain a schedule with makespan \makespanconstant T and activation cost
\costconstant A, for any . We also consider assignment costs for
jobs as in the generalized assignment problem, and using our framework, provide
algorithms that minimize the machine activation and the assignment cost
simultaneously. In addition, we present a greedy algorithm which only works for
the basic version and yields a makespan of and an activation cost .
For the uniformly related parallel machine scheduling problem, we develop a
polynomial time approximation scheme that outputs a schedule with the property
that the activation cost of the subset of machines is at most and the
makespan is at most for any
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
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