865 research outputs found
Directed Multicut is W[1]-hard, Even for Four Terminal Pairs
We prove that Multicut in directed graphs, parameterized by the size of the
cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if
restricted to instances with only four terminal pairs. This negative result
almost completely resolves one of the central open problems in the area of
parameterized complexity of graph separation problems, posted originally by
Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case
of three terminal pairs open.
Our gadget methodology allows us also to prove W[1]-hardness of the Steiner
Orientation problem parameterized by the number of terminal pairs, resolving an
open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math.
27(3):1503-1513 (2013)].Comment: v2: Added almost tight ETH lower bound
Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints
We study the parameterized problem of satisfying ``almost all'' constraints
of a given formula over a fixed, finite Boolean constraint language
, with or without weights. More precisely, for each finite Boolean
constraint language , we consider the following two problems. In Min
SAT, the input is a formula over and an integer , and
the task is to find an assignment that
satisfies all but at most constraints of , or determine that no such
assignment exists. In Weighted Min SAT), the input additionally
contains a weight function and an integer ,
and the task is to find an assignment such that (1) satisfies
all but at most constraints of , and (2) the total weight of the
violated constraints is at most . We give a complete dichotomy for the
fixed-parameter tractability of these problems: We show that for every Boolean
constraint language , either Weighted Min SAT is FPT; or
Weighted Min SAT is W[1]-hard but Min SAT is FPT; or Min
SAT is W[1]-hard. This generalizes recent work of Kim et al. (SODA
2021) which did not consider weighted problems, and only considered languages
that cannot express implications (as is used to, e.g.,
model digraph cut problems). Our result generalizes and subsumes multiple
previous results, including the FPT algorithms for Weighted Almost 2-SAT,
weighted and unweighted -Chain SAT, and Coupled Min-Cut, as well as
weighted and directed versions of the latter. The main tool used in our
algorithms is the recently developed method of directed flow-augmentation (Kim
et al., STOC 2022)
Solving hard cut problems via flow-augmentation
We present a new technique for designing FPT algorithms for graph cut
problems in undirected graphs, which we call flow augmentation. Our technique
is applicable to problems that can be phrased as a search for an (edge)
-cut of cardinality at most in an undirected graph with
designated terminals and .
More precisely, we consider problems where an (unknown) solution is a set of size at most such that (1) in , and are in
distinct connected components, (2) every edge of connects two distinct
connected components of , and (3) if we define the set as these edges for which there exists an -path with
, then separates from . We prove that
in this scenario one can in randomized time add a
number of edges to the graph so that with probability no
added edge connects two components of and becomes a minimum cut
between and .
We apply our method to obtain a randomized FPT algorithm for a notorious
"hard nut" graph cut problem we call Coupled Min-Cut. This problem emerges out
of the study of FPT algorithms for Min CSP problems, and was unamenable to
other techniques for parameterized algorithms in graph cut problems, such as
Randomized Contractions, Treewidth Reduction or Shadow Removal.
To demonstrate the power of the approach, we consider more generally Min
SAT(), parameterized by the solution cost. We show that every problem
Min SAT() is either (1) FPT, (2) W[1]-hard, or (3) able to express the
soft constraint , and thereby also the min-cut problem in directed
graphs. All the W[1]-hard cases were known or immediate, and the main new
result is an FPT algorithm for a generalization of Coupled Min-Cut
Flow-augmentation III: complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints
Randomized contractions meet lean decompositions
We show an algorithm that, given an -vertex graph and a parameter ,
in time finds a tree decomposition of with the
following properties:
* every adhesion of the tree decomposition is of size at most , and
* every bag of the tree decomposition is -unbreakable in for every
.
Here, a set is -unbreakable in if for every
separation of order at most in , we have or
. The resulting tree decomposition has arguably best
possible adhesion size boundsand unbreakability guarantees. Furthermore, the
parametric factor in the running time bound is significantly smaller than in
previous similar constructions. These improvements allow us to present
parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner
Multicut with improved parameteric factor in the running time bound.
The main technical insight is to adapt the notion of lean decompositions of
Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to
the parameterized setting.Comment: v2: New co-author (Magnus) and improved results on vertex
unbreakability of bags, v3: final changes, including new abstrac
Spatial Data on different scales in GEM:Allows for Quatification of Geomorphological changes following Extreme Events
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