14 research outputs found
Kernelization of Cycle Packing with Relaxed Disjointness Constraints
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1
On r-Simple k-Path and Related Problems Parameterized by k/r
Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems.
In the -Simple -Path problem, given a digraph on vertices and
integers , decide whether has an -simple -path, which is a walk
where every vertex occurs at most times and the total number of vertex
occurrences is . In the -Monomial Detection problem, given an
arithmetic circuit that encodes some polynomial on variables and
integers , decide whether has a monomial of degree where the
degree of each variable is at most~. In the -Set -Packing problem,
given a universe , positive integers , and a collection of
sets of size whose elements belong to , decide whether there exists a
subcollection of of size where each element occurs in
at most sets of . Abasi et al. and Gabizon et al. proved that
the three problems are single-exponentially fixed-parameter tractable (FPT)
when parameterized by , where for -Set -Packing
and asked whether the factor in the exponent can be avoided.
We consider their question from a wider perspective: are the above problems
FPT when parameterized by only? We resolve the wider question by (a)
obtaining a -time algorithm for
-Simple -Path on digraphs and a -time
algorithm for -Simple -Path on undirected graphs (i.e., for undirected
graphs we answer the original question in affirmative), (b) showing that
-Set -Packing is FPT, and (c) proving that -Monomial Detection
is para-NP-hard. For -Set -Packing, we obtain a polynomial kernel for
any fixed , which resolves a question posed by Gabizon et al. regarding the
existence of polynomial kernels for problems with relaxed disjointness
constraints
Combining Shortest Paths, Bottleneck Paths and Matrix Multiplication
We provide a formal mathematical definition of the Shortest Paths for All Flows (SP-AF) problem and provide many efficient algorithms. The SP-AF problem combines the well known Shortest Paths (SP) and Bottleneck Paths (BP) problems, and can be solved by utilising matrix multiplication. Thus in our research of the SP-AF problem, we also make a series of contributions to the underlying topics of the SP problem, the BP problem, and matrix multiplication.
For the topic of matrix multiplication we show that on an n-by-n two dimensional (2D) square mesh array, two n-by-n matrices can be multiplied in exactly 1.5n â 1 communication steps. This halves the number of communication steps required by the well known Cannonâs algorithm that runs
on the same sized mesh array.
We provide two contributions for the SP problem. Firstly, we enhance the breakthrough algorithm by Alon, Galil and Margalit (AGM), which was the first algorithm to achieve a deeply sub-cubic time bound for solving the All Pairs Shortest Paths (APSP) problem on dense directed graphs. Our enhancement allows the algorithm by AGM to remain sub-cubic for larger upper bounds on integer edge costs. Secondly, we show that for graphs with n vertices, the APSP problem can be solved in exactly 3n â 2 communication steps on an n-by-n 2D square mesh array. This improves on the previous result of 3.5n communication steps achieved by Takaoka and Umehara.
For the BP problem, we show that we can compute the bottleneck of the entire graph without solving the All Pairs Bottleneck Paths (APBP) problem, resulting in a much more efficient time bound.
Finally we define an algebraic structure called the distance/flow semi-ring to formally introduce the SP-AF problem, and we provide many algorithms for solving the Single Source SP-AF (SSSP-AF) problem and the All Pairs SP-AF (APSP-AF) problem. For the APSP-AF problem, algebraic algorithms are given that utilise faster matrix multiplication over a ring
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum