353 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Fast Algorithms for Separable Linear Programs
In numerical linear algebra, considerable effort has been devoted to
obtaining faster algorithms for linear systems whose underlying matrices
exhibit structural properties. A prominent success story is the method of
generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices.
On the other hand, the majority of recent developments in the design of
efficient linear program (LP) solves do not leverage the ideas underlying these
faster linear system solvers nor consider the separable structure of the
constraint matrix.
We give a faster algorithm for separable linear programs. Specifically, we
consider LPs of the form , where the
graphical support of the constraint matrix is -separable. These include flow problems on planar graphs
and low treewidth matrices among others. We present an time algorithm for these LPs, where is
the relative accuracy of the solution.
Our new solver has two important implications: for the -multicommodity
flow problem on planar graphs, we obtain an algorithm running in
time in the high accuracy regime; and when the
support of is -separable with , our
algorithm runs in time, which is nearly optimal. The latter
significantly improves upon the natural approach of combining interior point
methods and nested dissection, whose time complexity is lower bounded by
, where is the
matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we
recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data
structure machinery.Comment: 55 pages. To appear at SODA 202
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Dynamic programming on bipartite tree decompositions
We revisit a graph width parameter that we dub bipartite treewidth, along
with its associated graph decomposition that we call bipartite tree
decomposition. Bipartite treewidth can be seen as a common generalization of
treewidth and the odd cycle transversal number. Intuitively, a bipartite tree
decomposition is a tree decomposition whose bags induce almost bipartite graphs
and whose adhesions contain at most one vertex from the bipartite part of any
other bag, while the width of such decomposition measures how far the bags are
from being bipartite. Adapted from a tree decomposition originally defined by
Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by
Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial
role for solving problems related to odd-minors, which have recently attracted
considerable attention. As a first step toward a theory for solving these
problems efficiently, the main goal of this paper is to develop dynamic
programming techniques to solve problems on graphs of small bipartite
treewidth. For such graphs, we provide a number of para-NP-completeness
results, FPT-algorithms, and XP-algorithms, as well as several open problems.
In particular, we show that -Subgraph-Cover, Weighted Vertex
Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are
parameterized by bipartite treewidth. We provide the following complexity
dichotomy when is a 2-connected graph, for each of -Subgraph-Packing,
-Induced-Packing, -Scattered-Packing, and -Odd-Minor-Packing problem:
if is bipartite, then the problem is para-NP-complete parameterized by
bipartite treewidth while, if is non-bipartite, then it is solvable in
XP-time. We define 1--treewidth by replacing the bipartite graph
class by any class . Most of the technology developed here works for
this more general parameter.Comment: Presented in IPEC 202
How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach
In this paper we study a resource allocation problem that encodes correlation
between items in terms of \conflict and maximizes the minimum utility of the
agents under a conflict free allocation. Admittedly, the problem is
computationally hard even under stringent restrictions because it encodes a
variant of the {\sc Maximum Weight Independent Set} problem which is one of the
canonical hard problems in both classical and parameterized complexity.
Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from
the classical complexity perspective to draw the boundary between {\sf
NP}-hardness and tractability for a constant number of agents. The problem was
shown to be hard even for small constant number of agents and various other
restrictions on the underlying graph. Notwithstanding this computational
barrier, we notice that there are several parameters that are worth studying:
number of agents, number of items, combinatorial structure that defines the
conflict among the items, all of which could well be small under specific
circumstancs. Our search rules out several parameters (even when taken
together) and takes us towards a characterization of families of input
instances that are amenable to polynomial time algorithms when the parameters
are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)}
algorithm for our problem where denotes the number of items that
significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using
ideas from FFT based fast polynomial multiplication; and we identify simple
graph classes relevant to our problem's motivation that admit efficient
algorithms
Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor
We prove that for and an absolute constant , if and is a random subset of where
each is included in independently with probability for
each , then asymptotically almost surely there is an order-
Latin square in which the entry in the th row and th column lies in
. The problem of determining the threshold probability for the
existence of an order- Latin square was raised independently by Johansson,
by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides
an upper bound which is tight up to a factor of and strengthens the
bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous
results for Steiner triple systems and -factorizations of complete graphs,
and moreover, we show that each of these thresholds is at most the threshold
for the existence of a -factorization of a nearly complete regular bipartite
graph.Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the
AM
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
- …