597 research outputs found
Span programs and quantum algorithms for st-connectivity and claw detection
We introduce a span program that decides st-connectivity, and generalize the
span program to develop quantum algorithms for several graph problems. First,
we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries
to the n x n adjacency matrix to decide if vertices s and t are connected,
under the promise that they either are connected by a path of length at most d,
or are disconnected. We also show that if T is a path, a star with two
subdivided legs, or a subdivision of a claw, its presence as a subgraph in the
input graph G can be detected with O(n) quantum queries to the adjacency
matrix. Under the promise that G either contains T as a subgraph or does not
contain T as a minor, we give O(n)-query quantum algorithms for detecting T
either a triangle or a subdivision of a star. All these algorithms can be
implemented time efficiently and, except for the triangle-detection algorithm,
in logarithmic space. One of the main techniques is to modify the
st-connectivity span program to drop along the way "breadcrumbs," which must be
retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure
Graph editing to a fixed target
For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H
Graph editing to a fixed target
For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
Detecting and counting small subgraphs, and evaluating a parameterized Tutte polynomial: lower bounds via toroidal grids and Cayley graph expanders
Given a graph property , we consider the problem , where the input is a pair of a graph and a positive integer , and the task is to decide whether contains a -edge subgraph that satisfies . Specifically, we study the parameterized complexity of and of its counting problem with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties : the decision problem always admits an FPT algorithm and the counting problem always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property which, if satisfied, yields fixed-parameter tractability and otherwise -hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for that run in time for any computable function . As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of . This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial of a graph , to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, corresponds to the number of -forests in the graph . Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of at every pair of rational coordinates
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Minimal Conflicting Sets for the Consecutive Ones Property in ancestral genome reconstruction
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be
ordered in such a way that all 1's on each row are consecutive. A Minimal
Conflicting Set is a set of rows that does not have the C1P, but every proper
subset has the C1P. Such submatrices have been considered in comparative
genomics applications, but very little is known about their combinatorial
structure and efficient algorithms to compute them. We first describe an
algorithm that detects rows that belong to Minimal Conflicting Sets. This
algorithm has a polynomial time complexity when the number of 1's in each row
of the considered matrix is bounded by a constant. Next, we show that the
problem of computing all Minimal Conflicting Sets can be reduced to the joint
generation of all minimal true clauses and maximal false clauses for some
monotone boolean function. We use these methods on simulated data related to
ancestral genome reconstruction to show that computing Minimal Conflicting Set
is useful in discriminating between true positive and false positive ancestral
syntenies. We also study a dataset of yeast genomes and address the reliability
of an ancestral genome proposal of the Saccahromycetaceae yeasts.Comment: 20 pages, 3 figure
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