160 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Improved Hardness of Approximating k-Clique under ETH
In this paper, we prove that assuming the exponential time hypothesis (ETH),
there is no -time algorithm that can decide
whether an -vertex graph contains a clique of size or contains no clique
of size , and no FPT algorithm can decide whether an input graph has a
clique of size or no clique of size , where is some function
in . Our results significantly improve the previous works [Lin21,
LRSW22]. The crux of our proof is a framework to construct gap-producing
reductions for the -Clique problem. More precisely, we show that given an
error-correcting code that is locally testable
and smooth locally decodable in the parallel setting, one can construct a
reduction which on input a graph outputs a graph in time such that:
If has a clique of size , then has a clique of size
, where .
If has no clique of size , then has no clique of size
for some constant .
We then construct such a code with and
, establishing the hardness results above.
Our code generalizes the derivative code [WY07] into the case with a super
constant order of derivatives.Comment: 48 page
Fully dynamic approximation schemes on planar and apex-minor-free graphs
The classic technique of Baker [J. ACM '94] is the most fundamental approach
for designing approximation schemes on planar, or more generally
topologically-constrained graphs, and it has been applied in a myriad of
different variants and settings throughout the last 30 years. In this work we
propose a dynamic variant of Baker's technique, where instead of finding an
approximate solution in a given static graph, the task is to design a data
structure for maintaining an approximate solution in a fully dynamic graph,
that is, a graph that is changing over time by edge deletions and edge
insertions. Specifically, we address the two most basic problems -- Maximum
Weight Independent Set and Minimum Weight Dominating Set -- and we prove the
following: for a fully dynamic -vertex planar graph , one can:
* maintain a -approximation of the maximum weight of an
independent set in with amortized update time ; and,
* under the additional assumption that the maximum degree of the graph is
bounded at all times by a constant, also maintain a
-approximation of the minimum weight of a dominating set in
with amortized update time .
In both cases, is doubly-exponential in
and the data structure can be initialized in
time . All our results in fact hold in the
larger generality of any graph class that excludes a fixed apex-graph as a
minor.Comment: 37 pages, accepted to SODA '2
Parameterized Complexity Classification for Interval Constraints
Constraint satisfaction problems form a nicely behaved class of problems that
lends itself to complexity classification results. From the point of view of
parameterized complexity, a natural task is to classify the parameterized
complexity of MinCSP problems parameterized by the number of unsatisfied
constraints. In other words, we ask whether we can delete at most
constraints, where is the parameter, to get a satisfiable instance. In this
work, we take a step towards classifying the parameterized complexity for an
important infinite-domain CSP: Allen's interval algebra (IA). This CSP has
closed intervals with rational endpoints as domain values and employs a set
of 13 basic comparison relations such as ``precedes'' or ``during'' for
relating intervals. IA is a highly influential and well-studied formalism
within AI and qualitative reasoning that has numerous applications in, for
instance, planning, natural language processing and molecular biology. We
provide an FPT vs. W[1]-hard dichotomy for MinCSP for all . IA is sometimes extended with unions of the relations in or
first-order definable relations over , but extending our results to these
cases would require first solving the parameterized complexity of Directed
Symmetric Multicut, which is a notorious open problem. Already in this limited
setting, we uncover connections to new variants of graph cut and separation
problems. This includes hardness proofs for simultaneous cuts or feedback arc
set problems in directed graphs, as well as new tractable cases with algorithms
based on the recently introduced flow augmentation technique. Given the
intractability of MinCSP in general, we then consider (parameterized)
approximation algorithms and present a factor- fpt-approximation algorithm
Baby PIH: Parameterized Inapproximability of Min CSP
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP
theorem in the world of parameterized complexity. It asserts that no FPT
algorithm can distinguish a satisfiable 2CSP instance from one which is only
-satisfiable (where the parameter is the number of variables)
for some constant .
We consider a minimization version of CSPs (Min-CSP), where one may assign
values to each variable, and the goal is to ensure that every constraint is
satisfied by some choice among the pairs of values assigned to its
variables (call such a CSP instance -list-satisfiable). We prove the
following strong parameterized inapproximability for Min CSP: For every , it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even
-list-satisfiable. We refer to this statement as "Baby PIH", following the
recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the
combinatorial arguments underlying the Baby PCP theorem, overcoming some basic
obstacles that arise in the parameterized setting. Furthermore, our reduction
runs in time polynomially bounded in both the number of variables and the
alphabet size, and thus implies the Baby PCP theorem as well
On streaming approximation algorithms for constraint satisfaction problems
In this thesis, we explore streaming algorithms for approximating constraint
satisfaction problems (CSPs). The setup is roughly the following: A computer
has limited memory space, sees a long "stream" of local constraints on a set of
variables, and tries to estimate how many of the constraints may be
simultaneously satisfied. The past ten years have seen a number of works in
this area, and this thesis includes both expository material and novel
contributions. Throughout, we emphasize connections to the broader theories of
CSPs, approximability, and streaming models, and highlight interesting open
problems.
The first part of our thesis is expository: We present aspects of previous
works that completely characterize the approximability of specific CSPs like
Max-Cut and Max-Dicut with -space streaming algorithm (on
-variable instances), while characterizing the approximability of all CSPs
in space in the special case of "composable" (i.e., sketching)
algorithms, and of a particular subclass of CSPs with linear-space streaming
algorithms.
In the second part of the thesis, we present two of our own joint works. We
begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove
linear-space streaming approximation-resistance for all ordering CSPs (OCSPs),
which are "CSP-like" problems maximizing over sets of permutations. Next, we
present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and
Santhoshini Velusamy in which we investigate the -space streaming
approximability of symmetric Boolean CSPs with negations. We give explicit
-space sketching approximability ratios for several families of CSPs,
including Max-AND; develop simpler optimal sketching approximation
algorithms for threshold predicates; and show that previous lower bounds fail
to characterize the -space streaming approximability of Max-AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract
shortened for arXiv; formatted with Dissertate template (feel free to copy!);
exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX
2022
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