448 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Planar Disjoint Paths, Treewidth, and Kernels
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . We
study the problem through the lens of kernelization, aiming at efficiently
reducing the input size in terms of a parameter. We show that Planar Disjoint
Paths does not admit a polynomial kernel when parameterized by unless coNP
NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e},
Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel
unless the WK-hierarchy collapses. Our reduction carries over to the setting of
edge-disjoint paths, where the kernelization status remained open even in
general graphs.
On the positive side, we present a polynomial kernel for Planar Disjoint
Paths parameterized by , where denotes the treewidth of the input
graph. As a consequence of both our results, we rule out the possibility of a
polynomial-time (Turing) treewidth reduction to under the same
assumptions. To the best of our knowledge, this is the first hardness result of
this kind. Finally, combining our kernel with the known techniques [Adler,
Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver,
SICOMP'94] yields an alternative (and arguably simpler) proof that Planar
Disjoint Paths can be solved in time , matching the
result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure
On Conflict-Free Cuts: Algorithms and Complexity
One way to define the Matching Cut problem is: Given a graph , is there an
edge-cut of such that is an independent set in the line graph of
? We propose the more general Conflict-Free Cut problem: Together with the
graph , we are given a so-called conflict graph on the edges of
, and we ask for an edge-cutset of that is independent in .
Since conflict-free settings are popular generalizations of classical
optimization problems and Conflict-Free Cut was not considered in the
literature so far, we start the study of the problem. We show that the problem
is -complete even when the maximum degree of is 5 and
is 1-regular. The same reduction implies an exponential lower bound
on the solvability based on the Exponential Time Hypothesis. We also give
parameterized complexity results: We show that the problem is fixed-parameter
tractable with the vertex cover number of as a parameter, and we show
-hardness even when has a feedback vertex set of size one,
and the clique cover number of is the parameter. Since the clique
cover number of is an upper bound on the independence number of
and thus the solution size, this implies -hardness
when parameterized by the cut size. We list polynomial-time solvable cases and
interesting open problems. At last, we draw a connection to a symmetric variant
of SAT.Comment: 13 pages, 3 figure
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A Graph-Theoretic Formulation of Exploratory Blockmodeling
We present a new simple graph-theoretic formulation of the exploratory blockmodeling problem on undirected and unweighted one-mode networks. Our formulation takes as input the network G and the maximum number t of blocks for the solution model. The task is to find a minimum-size set of edge insertions and deletions that transform the input graph G into a graph G\u27 with at most t neighborhood classes. Herein, a neighborhood class is a maximal set of vertices with the same neighborhood. The neighborhood classes of G\u27 directly give the blocks and block interactions of the computed blockmodel.
We analyze the classic and parameterized complexity of the exploratory blockmodeling problem, provide a branch-and-bound algorithm, an ILP formulation and several heuristics. Finally, we compare our exact algorithms to previous ILP-based approaches and show that the new algorithms are faster for t ? 4
Parameterized Matroid-Constrained Maximum Coverage
In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended.
We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid ? = (V, ?) of rank k on a ground set V and a coverage function f on V, the goal is to find an independent set S ? ? maximizing f(S). This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum k-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency ? (i.e., any element of the underlying universe of the coverage function appears in at most ? sets), we design a procedure, parameterized by some integer ?, to extract in polynomial time an approximate kernel of size ? ? k that is guaranteed to contain a 1 - (? - 1)/? approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a 1 - ? approximation in time (?/?)^O(k) ? |V|^O(1). This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, as the kernel has a very simple characterization, it can be constructed in the streaming setting
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