173 research outputs found
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
On the Parameterized Complexity of Deletion to ?-Free Strong Components
Directed Feedback Vertex Set (DFVS) is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the H-SCC Deletion problem. Here, one is given a digraph D, an integer k and the objective is to decide whether there is a vertex set of size at most k whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family H as (not necessarily induced) subgraphs. When H comprises only the digraph with a single arc, then this problem is precisely DFVS.
Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if H only contains rooted graphs or if H contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of Göke et al. [CIAC 2019] for the 1-Out-Regular Vertex Deletion and Bounded Size Strong Component Vertex Deletion problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for DFVS, without using the heavy machinery of shadow removal as is done by Göke et al. [CIAC 2019].publishedVersio
A Polynomial Kernel for Funnel Arc Deletion Set
In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider ?-Arc Deletion Set (?-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class ?. In this work, we choose ? to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernel for this problem was known. Our main result is a kernel for Funnel-ADS with ?(k?) many vertices and ?(k?) many arcs, computable in ?(nm) time, where n is the number of vertices and m the number of arcs of the input digraph
Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs
We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with O?(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in O?(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in O?(n?m) worst-case time and queries in O(?m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008]
Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set
The Directed Feedback Vertex Set (DFVS) problem takes as input a directed
graph~ and seeks a smallest vertex set~ that hits all cycles in . This
is one of Karp's 21 -complete problems. Resolving the
parameterized complexity status of DFVS was a long-standing open problem until
Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability
via a -time algorithm, where .
Here we show fixed-parameter tractability of two generalizations of DFVS:
- Find a smallest vertex set such that every strong component of
has size at most~: we give an algorithm solving this problem in time
. This generalizes an algorithm by Xiao
[JCSS 2017] for the undirected version of the problem.
- Find a smallest vertex set such that every non-trivial strong component
of is 1-out-regular: we give an algorithm solving this problem in time
.
We also solve the corresponding arc versions of these problems by
fixed-parameter algorithms
All solution graphs in multidimensional screening
We study general discrete-types multidimensional screening without any noticeable restrictions on valuations, using instead epsilon-relaxation of the incentive-compatibility constraints. Any active (becoming equality) constraint can be perceived as "envy" arc from one type to another, so the set of active constraints is a digraph. We find that: (1) any solution has an in-rooted acyclic graph ("river"); (2) for any logically feasible river there exists a screening problem resulting in such river. Using these results, any solution is characterized both through its spanning-tree and through its Lagrange multipliers, that can help in finding solutions and their efficiency/distortion properties.incentive compatibility; multidimensional screening; second-degree price discrimination; non-linear pricing; graphs
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
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