849 research outputs found
Packing odd -joins with at most two terminals
Take a graph , an edge subset , and a set of
terminals where is even. The triple is
called a signed graft. A -join is odd if it contains an odd number of edges
from . Let be the maximum number of edge-disjoint odd -joins.
A signature is a set of the form where and is even. Let be the minimum cardinality a -cut
or a signature can achieve. Then and we say that
packs if equality holds here.
We prove that packs if the signed graft is Eulerian and it
excludes two special non-packing minors. Our result confirms the Cycling
Conjecture for the class of clutters of odd -joins with at most two
terminals. Corollaries of this result include, the characterizations of weakly
and evenly bipartite graphs, packing two-commodity paths, packing -joins
with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title
"the cycling property for the clutter of odd st-walks"
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
Clean clutters and dyadic fractional packings
A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas
Odd Paths, Cycles and -joins: Connections and Algorithms
Minimizing the weight of an edge set satisfying parity constraints is a
challenging branch of combinatorial optimization as witnessed by the binary
hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization''
(Chapter 80). This area contains relevant graph theory problems including open
cases of the Max Cut problem, or some multiflow problems. We clarify the
interconnections of some problems and establish three levels of difficulties.
On the one hand, we prove that the Shortest Odd Path problem in an undirected
graph without cycles of negative total weight and several related problems are
NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem
27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we
provide a polynomial-time algorithm to the closely related and well-studied
Minimum-weight Odd -Join problem for non-negative weights, whose
complexity, however, was not known; more generally, we solve the Minimum-weight
Odd -Join problem in FPT time when parameterized by . If negative
weights are also allowed, then finding a minimum-weight odd -join is
equivalent to the Minimum-weight Odd -Join problem for arbitrary weights,
whose complexity is only conjectured to be polynomially solvable. The analogous
problems for digraphs are also considered.Comment: 24 pages, 2 figure
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
FPT algorithms for path-transversal and cycle-transversal problems
AbstractWe study the parameterized complexity of several vertex- and edge-deletion problems on graphs, parameterized by the number p of deletions. The first kind of problems are separation problems on undirected graphs, where we aim at separating distinguished vertices in a graph. The second kind of problems are feedback set problems on group-labelled graphs, where we aim at breaking nonnull cycles in a graph having its arcs labelled by elements of a group. We obtain new FPT algorithms for these different problems, relying on a generic O∗(4p) algorithm for breaking paths of a homogeneous path system
Ideal Clutters
Let E be a finite set of elements, and let C be a family of subsets of E called members. We say that C is a clutter over ground set E if no member is contained in another. The clutter C is ideal if every extreme point of the polyhedron { x>=0 : x(C) >= 1 for every member C } is integral.
Ideal clutters are central objects in Combinatorial Optimization, and they have deep connections to several other areas. To integer programmers, they are the underlying structure of set covering integer programs that are easily solvable. To graph theorists, they are manifest in the famous theorems of Edmonds and Johnson on T-joins, of Lucchesi and Younger on dijoins, and of Guenin on the characterization of weakly bipartite graphs; not to mention they are also the set covering analogue of perfect graphs. To matroid theorists, they are abstractions of Seymour’s sums of circuits property as well as his f-flowing property. And finally, to combinatorial optimizers, ideal clutters host many minimax theorems and are extensions of totally unimodular and balanced matrices.
This thesis embarks on a mission to develop the theory of general ideal clutters.
In the first half of the thesis, we introduce and/or study tools for finding deltas, extended odd holes and their blockers as minors; identically self-blocking clutters; exclusive, coexclusive and opposite pairs; ideal minimally non-packing clutters and the τ = 2 Conjecture; cuboids; cube-idealness; strict polarity; resistance; the sums of circuits property; and minimally non-ideal binary clutters and the f-Flowing Conjecture.
While the first half of the thesis includes many broad and high-level contributions that are accessible to a non-expert reader, the second half contains three deep and technical contributions, namely, a character- ization of an infinite family of ideal minimally non-packing clutters, a structure theorem for ±1-resistant sets, and a characterization of the minimally non-ideal binary clutters with a member of cardinality three.
In addition to developing the theory of ideal clutters, a main goal of the thesis is to trigger further research on ideal clutters. We hope to have achieved this by introducing a handful of new and exciting conjectures on ideal clutters
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