Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

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Browsing Combinatorics and Optimization by Author "Guenin, Bertrand"

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    Comparing Intersection Cut Closures using Simple Families of Lattice-Free Convex Sets
    (University of Waterloo, 2022-04-26) Stuive, Leanne; Guenin, Bertrand; Tuncel, Levent
    Mixed integer programs are a powerful mathematical tool, providing a general model for expressing both theoretically difficult and practically useful problems. One important subroutine of algorithms solving mixed integer programs is a cut generation procedure. The job of a cut generation procedure is to produce a linear inequality that separates a given infeasible point x* (usually a basic feasible solution of the linear programming relaxation) from the set of feasible solutions for the problem at hand. Early and well-known cut generation procedures rely on analyzing a single row of the simplex tableau for x*. Andersen et al. renewed interest in d-row cuts (i.e. cuts derived from d rows of the simplex tableau) by showing that these cuts afford some theoretical benefit. One lens from which to study d-row cuts is in the context of the intersection cuts of Balas and, in particular, intersection cuts obtained from lattice-free convex sets. The strongest d-row intersection cuts are obtained from maximal lattice-free convex sets in $R^d$ - all of which are polyhedra with at most $2^d$ facets. This thesis is concerned with theoretical comparison of the d-row cuts generated by different families of maximal lattice-free convex sets. We use the gauge measure to appraise the quality of the approximation. The main area of focus is 2-row cuts. The problem of generating 2-row cuts can be re-posed as generating valid inequalities for a mixed integer linear set F with two free integer variables and any number of non-negative continuous variables, where there are two defining equations. Every minimal valid inequality for the convex hull of F is an intersection cut generated by a maximal lattice-free split, triangle or quadrilateral. The family of maximal lattice-free triangles can be subdivided into the families of type 1, type 2, and type 3 triangles. Previous results of Basu et al. and Awate et al. compare how well the inequalities from one of these families approximates the convex hull of F (a.k.a. the corner polyhedron). In particular, the closure of all type 2 triangle inequalities is shown to be within a factor of 3/2 of the corner polyhedron. The authors also provide an instance where all type 2 triangles inequalities cannot approximate the corner polyhedron better than a factor of 9/8. The same bounds are shown for type 3 triangles and quadrilaterals. These results are obtained not by directly comparing the given closures to the convex hull of F, but rather to each other. In this thesis, we tighten one of the underlying bounds, showing that the closure of all type 2 triangle inequalities are within a factor of 5/4 of the closure of all quadrilateral inequalities. We also consider the sub-family of quadrilaterals where opposite edges have equal slope. We show that these parallelogram cuts can be approximated by all type 2 triangle inequalities within a factor of 9/8 and there exist instances where no better approximation is possible. In proving both these bounds, we use a subset of the family of type 2 triangles; we call the members of this sub-family ray-sliding triangles. A secondary area of focus in this thesis is d-row cuts for d >= 3. For d-row cuts in general, the underlying maximal lattice-free convex sets in $R^d$ are not easily classified. Absent a classification, Averkov et al, show that all inequalities generated by lattice-free convex sets with at most $i$ facets approximate the corner polyhedron within a finite factor only when $i > 2^{d-1}$. Here we take a different tact and try to prove analogues of 2-row cut results. We extend the proof techniques to obtain a constant factor approximation between two structured families of maximal lattice-free convex sets in $R^d$ for d >= 3.
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    A Generalization to Signed Graphs of a Theorem of Sergey Norin and Robin Thomas
    (University of Waterloo, 2019-12-19) Horrocks, Courtney; Guenin, Bertrand
    In this thesis we characterize the minimal non-planar extensions of a signed graph. We consider the following question: Given a subdivision of a planar signed graph (G, Σ), what are the minimal structures that can be added to the subdivision to make it non-planar? Sergey Norin and Robin Thomas answered this question for unsigned graphs, assuming almost 4-connectivity for G and H. By adapting their proof to signed graphs, we prove a generalization of their result.
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    Ideal Clutters
    (University of Waterloo, 2018-04-24) Abdi, Ahmad; Guenin, Bertrand
    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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    Recognizing Even-Cycle and Even-Cut Matroids
    (University of Waterloo, 2016-04-27) Heo, Cheolwon; Guenin, Bertrand
    Even-cycle and even-cut matroids are classes of binary matroids that generalize respectively graphic and cographic matroids. We give algorithms to check membership for these classes of matroids. We assume that the matroids are 3-connected and are given by their (0,1)-matrix representations. We first give an algorithm to check membership for p-cographic matroids that is a subclass of even-cut matroids. We use this algorithm to construct algorithms for membership problems for even-cycle and even-cut matroids and the running time of these algorithms is polynomial in the size of the matrix representations. However, we will outline only how theoretical results can be used to develop polynomial time algorithms and omit the details of algorithms.
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    Relaxations of the Maximum Flow Minimum Cut Property for Ideal Clutters
    (University of Waterloo, 2021-01-29) Ferchiou, Zouhaier; Guenin, Bertrand; Tuncel, Levent
    Given a family of sets, a covering problem consists of finding a minimum cost collection of elements that hits every set. This objective can always be bound by the maximum number of disjoint sets in the family, we refer to this as the covering dual, since when we allow covers to be fractional and relax the notion of disjoint sets, the natural Linear Programming (LP) formulations become duals and the optimal objective values of the two LPs match. A consequence of the Edmonds-Giles theorem about Totally Dual Integral systems is that if we assume the covering dual always has an optimal integer solution for every cost function, then we can always find an optimal integer cover. The converse does not hold in general, but a still standing conjecture from the mid-1970s states that the existence of an optimal integer cover for every cost function implies the existence of a 1/4-integer optimal solution to the dual for every cost function. In this thesis we discuss weaker versions of the conjecture and build tools allowing us to approach them.
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    Representations of even-cycle and even-cut matroids
    (University of Waterloo, 2021-08-27) Heo, Cheolwon; Guenin, Bertrand
    In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. A graft is a pair $(G,T)$ where $G$ is a graph and $T$ is a subset of vertices of $G$ such that each component of $G$ contains an even number of vertices in $T$. Let $U$ be a subset of vertices of $G$ and let $D:= delta_G(U)$ be a cut of $G$. We say that $D$ is even in $(G, T)$ if $|U \cap T|$ is even. A matroid $M$ is an even-cut matroid if there exists a graft $(G,T)$ such that circuits of $M$ corresponds to inclusion-wise minimal non-empty even cuts of $(G,T)$.\\ This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations. (a) Isomorphism problem: what is the relationship between two representations? (b) Bounding the number of representations: how many representations can a matroid have? (c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists? These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two $4$-connected graphs $G_1$ and $G_2$ have a pair of signatures $(\Sigma_1, \Sigma_2)$ such that $(G_1, \Sigma_1)$ and $(G_2, \Sigma_2)$ represent the same even-cycle matroids. This also characterize when $G_1$ and $G_2$ have a pair of terminal sets $(T_1, T_2)$ such that $(G_1,T_1)$ and $(G_2,T_2)$ represent the same even-cut matroid. For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a $(0,1)$-matrix over the finite field $\gf(2)$. The time-complexity of these algorithms is polynomial in the size of the input matrix.