A perfect clone in an ordinal election (i.e., an election where the voters rank the candidates in a strict linear order) is a set of candidates that each voter ranks consecutively. We consider different relaxations of this notion: *independent* or *subelection clones* are sets of candidates that only some of the voters recognize as a perfect clone, whereas *approximate clones* are sets of candidates such that every voter ranks their members close to each other, but not necessarily consecutively. We establish the complexity of identifying such imperfect clones, and of partitioning the candidates into families of imperfect clones. We also study the parameterized complexity of these problems with respect to a set of natural parameters such as the number of voters, the size or the number of imperfect clones we are searching for, or their level of imperfection.

Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$.

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix $A$ and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of $A$, and when parameterized by the dual tree-depth and the entry complexity of $A$; both these parameterization imply that $A$ is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the $\ell_1$-norm of the Graver basis is bounded by a function of the maximum $\ell_1$-norm of a circuit of $A$. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix $A$ that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the $\ell_1$-norm of the Graver basis of the constraint matrix, when parameterized by the $\ell_1$-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with $n$ variables and a constraint matrix with dual tree-depth $d$ and largest entry $\Delta$ are solvable in time $g(d,\Delta){poly}(n)$ for some function $g$. However, the dual tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth, and thus does not reflect its geometric structure. We prove that the minimum dual tree-depth of a row-equivalent matrix is equal to the branch-depth of the matroid defined by the columns of the matrix. We design a fixed parameter algorithm for computing branch-depth of matroids represented over a finite field and a fixed parameter algorithm for computing a row-equivalent matrix with minimum dual tree-depth. Finally, we use these results to obtain an algorithm for integer programming running in time $g(d^*,\Delta){poly}(n)$ where $d^*$ is the branch-depth of the constraint matrix; the branch-depth cannot be replaced by the more permissive notion of branch-width.

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry Δ are solvable in time g(d,Δ) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and Δ. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.