arXiv subjects
Publications and source records attributed to Bertrand Guenin.
We design new tools to study variants of Total Dual Integrality. As an application, we obtain a geometric characterization of Total Dual Integrality for the case where the associated polyhedron is non-degenerate. We also give sufficient conditions for a system to be Totally Dual Dyadic, and prove new special cases of Seymour's Dyadic conjecture on ideal clutters.
A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.
A vector is \emph{dyadic} if each of its entries is a dyadic rational number, i.e. of the form $\frac{a}{2^k}$ for some integers $a,k$ with $k\geq 0$. A linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ for $w$ integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of \emph{dyadic generating sets for cones and subspaces}, the former being the dyadic analogue of \emph{Hilbert bases}, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the \emph{density} of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matrices, $T$-joins, cycles, and perfect matchings of a graph.
A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that, given any clutter $\mathcal{C}$ over $2n$ elements via a filter oracle, decides whether or not $\mathcal{C}$ is ideal. We prove that in the worst case, $\mathfrak{A}_{2n}$ must make at least $2^n$ calls to the filter oracle. Our proof uses the theory of cuboids.
Take a graph $G$, an edge subset $Σ\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,Σ,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from $Σ$. Let $ν$ be the maximum number of edge-disjoint odd $T$-joins. A signature is a set of the form $Σ\triangle δ(U)$ where $U\subseteq V(G)$ and $|U\cap T)$ is even. Let $τ$ be the minimum cardinality a $T$-cut or a signature can achieve. Then $ν\leq τ$ and we say that $(G,Σ,T)$ packs if equality holds here. We prove that $(G,Σ,T)$ 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 $T$-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing $T$-joins with at most four terminals, and a new result on covering edges with cuts.
A seminal result by Whitney describes when two graphs have the same cycles. We consider the analogous problem for even cycle matroids. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. We consider the problem of determining the relation between two signed graphs representing the same even cycle matroid. We refer to this problem as the Isomorphism Problem for even cycle matroids. We present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that, up to simple operations, any two signed graphs representing the same even cycle matroid are either in one of these classes, or related by a modification of an operation for graphic matroids, or belonging to a small set of examples.
We give an "excluded minor" and a "structural" characterization of digraphs D that have the property that for every subdigraph H of D, the maximum number of disjoint circuits in H is equal to the minimum cardinality of a subset T of V(H) such that H\T is acyclic.