9 research outputs found
Parameterized Complexity of Equality MinCSP
We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as ?, where the relations are defined via first-order formulas whose only predicate is =. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP(?) for every finite equality language ?, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the "cut requests" come as disjunctions over O(1) individual cut requests s_i ? t_i. We also consider singleton expansions of equality languages, enriching an equality language with the capability for assignment constraints (x = i) for either a finite or infinitely many constants i, and fully characterize the complexity of the resulting MinCSP
Finding Small Satisfying Assignments Faster Than Brute Force: {A} Fine-grained Perspective into {B}oolean Constraint Satisfaction
To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly . More precisely, we aim to determine, for any finite constraint family, the optimal running time required to find satisfying assignments that set precisely of the variables to . Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of into four regimes: (1) Brute force is essentially best-possible, i.e., , (2) the best algorithms are as fast as current -clique algorithms, i.e., , (3) the exponent has sublinear dependence on with , or (4) the problem is fixed-parameter tractable, i.e., . This yields a more fine-grained perspective than a previous FPT/W[1]-hardness dichotomy (Marx, Computational Complexity 2005). Our most interesting technical contribution is a -time algorithm for SubsetSum with precedence constraints parameterized by the target -- particularly the approach, based on generalizing a bound on the Frobenius coin problem to a setting with precedence constraints, might be of independent interest
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