38 research outputs found
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
Lagrangian Relaxation and Partial Cover
Lagrangian relaxation has been used extensively in the design of
approximation algorithms. This paper studies its strengths and limitations when
applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Lagrangian Relaxation and Partial Cover (Extended Abstract)
Lagrangian relaxation has been used extensively in the design of
approximation algorithms. This paper studies its strengths and
limitations when applied to Partial Cover.
We show that for Partial Cover in general no algorithm that uses
Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP)
-approximation as a black box can yield an approximation
factor better than~. This matches the upper bound
given by K"onemann et al. (ESA 2006, pages
468--479).
Faced with this limitation we study a specific, yet broad class of
covering problems: Partial Totally Balanced Cover. By carefully
analyzing the inner workings of the LMP algorithm we are able to
give an almost tight characterization of the integrality gap of the
standard linear relaxation of the problem. As a consequence we
obtain improved approximations for the Partial version of Multicut
and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with
-Blocks
Algorithms and Adaptivity Gaps for Stochastic k-TSP
Given a metric and a , the classic
\textsf{k-TSP} problem is to find a tour originating at the
of minimum length that visits at least nodes in . In this work,
motivated by applications where the input to an optimization problem is
uncertain, we study two stochastic versions of \textsf{k-TSP}.
In Stoch-Reward -TSP, originally defined by Ene-Nagarajan-Saket [ENS17],
each vertex in the given metric contains a stochastic reward .
The goal is to adaptively find a tour of minimum expected length that collects
at least reward ; here "adaptively" means our next decision may depend on
previous outcomes. Ene et al. give an -approximation adaptive
algorithm for this problem, and left open if there is an -approximation
algorithm. We totally resolve their open question and even give an
-approximation \emph{non-adaptive} algorithm for this problem.
We also introduce and obtain similar results for the Stoch-Cost -TSP
problem. In this problem each vertex has a stochastic cost , and the
goal is to visit and select at least vertices to minimize the expected
\emph{sum} of tour length and cost of selected vertices. This problem
generalizes the Price of Information framework [Singla18] from deterministic
probing costs to metric probing costs.
Our techniques are based on two crucial ideas: "repetitions" and "critical
scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for
our problems, if we truncate the random variables at an ideal threshold and
repeat, then their expected values form a good surrogate. Unfortunately, this
ideal threshold is adaptive as it depends on how far we are from achieving our
target , so we truncate at various different scales and identify a
"critical" scale.Comment: ITCS 202
Revisiting Garg's 2-Approximation Algorithm for the k-MST Problem in Graphs
This paper revisits the 2-approximation algorithm for -MST presented by
Garg in light of a recent paper of Paul et al.. In the -MST problem, the
goal is to return a tree spanning vertices of minimum total edge cost. Paul
et al. extend Garg's primal-dual subroutine to improve the approximation ratios
for the budgeted prize-collecting traveling salesman and minimum spanning tree
problems. We follow their algorithm and analysis to provide a cleaner version
of Garg's result. Additionally, we introduce the novel concept of a kernel
which allows an easier visualization of the stages of the algorithm and a
clearer understanding of the pruning phase. Other notable updates include
presenting a linear programming formulation of the -MST problem, including
pseudocode, replacing the coloring scheme used by Garg with the simpler concept
of neutral sets, and providing an explicit potential function.Comment: Proceedings of SIAM Symposium on Simplicity in Algorithms (SOSA) 202
Pruning 2-Connected Graphs
Given an edge-weighted undirected graph with a specified set of
terminals, let the emph{density} of any subgraph be the ratio of
its weight/cost to the number of terminals it contains. If is
2-connected, does it contain smaller 2-connected subgraphs of
density comparable to that of ? We answer this question in the
affirmative by giving an algorithm to emph{prune} and find such
subgraphs of any desired size, at the cost of only a logarithmic
increase in density (plus a small additive factor).
We apply the pruning techniques to give algorithms for two NP-Hard
problems on finding large 2-vertex-connected subgraphs of low cost;
no previous approximation algorithm was known for either problem. In
the kv problem, we are given an undirected graph with edge
costs and an integer ; the goal is to find a minimum-cost
2-vertex-connected subgraph of containing at least
vertices. In the bv problem, we are given the graph with edge
costs, and a budget ; the goal is to find a 2-vertex-connected
subgraph of with total edge cost at most that maximizes
the number of vertices in . We describe an
approximation for the kv problem, and a bicriteria approximation
for the bv problem that gives an
approximation, while violating the budget by a factor of at most