Delta minors, delta free clutters, and entanglement

For an integer n ≥ 3, the clutter ∆n := {1, 2}, {1, 3}, . . ., {1, n}, {2, 3, . . ., n} is called a delta of dimension n, whose members are the lines of a degenerate projective plane. In his seminal paper on nonideal clutters, Lehman revealed the role of the deltas as a distinct class of minimally nonideal clutters [The width length inequality and degenerate projective planes, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 1, AMS, Providence, RI, 1990, pp. 101-105]. A clutter is delta free if it has no delta minor. Binary clutters, ideal clutters, and clutters with the packing property are examples of delta free clutters. In this paper, we introduce and study basic geometric notions defined on clutters, including entanglement between clutters, a notion that is intimately linked with set covering polyhedra having a convex union. We will then investigate the surprising geometric attributes of delta minors and delta free clutters

Identically self-blocking clutters

A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter

Ideal clutters that do not pack

For a clutter over ground set E, a pair of distinct elements e, f ∈ E are coexclusive if every minimal cover contains at most one of them. An identification of is another clutter obtained after identifying coexclusive elements of . If a clutter is nonpacking, then so is any identification of it. Inspired by this observation, and impelled by the lack of a qualitative characterization for ideal minimally nonpacking (mnp) clutters, we reduce ideal mnp clutters even further by taking their identifications. In doing so, we reveal chains of ideal mnp clutters, demonstrate the centrality of mnp clutters with covering number two, as well as provide a qualitative characterization of irreducible ideal mnp clutters with covering number two. At the core of this characterization lies a class of objects, called marginal cuboids, that naturally give rise to ideal nonpacking clutters with covering number two. We present an explicit class of marginal cuboids, and show that the corresponding clutters have one of Q 6 , Q 2, 1 , Q 10 as a minor, where Q 6 , Q 2, 1 are known ideal mnp clutters, and Q 10 is a new ideal mnp clutter

Opposite elements in clutters

Let E be a finite set of elements, and let L be a clutter over ground set E. We say distinct elements e, f are opposite if every member and every minimal cover of L contains at most one of e, f. In this paper, we investigate opposite elements and reveal a rich theory underlying such a seemingly simple restriction. The clutter C obtained from L after identifying some opposite elements is called an identification of L; inversely, L is called a split of C. We will show that splitting preserves three clutter properties, i.e., idealness, the max-flow min-cut property, and the packing property. We will also display several natural examples in which a clutter does not have these properties but a split of them does. We will develop tools for recognizing when splitting is not a useful operation, and as well, we will characterize when identification preserves the three mentioned properties. We will also make connections to spanning arborescences, Steiner trees, comparability graphs, degenerate projective planes, binary clutters, matroids, as well as the results of Menger, Ford and Fulkerson, the Replication Conjecture, and a conjecture on ideal, minimally nonpacking clutters

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

Resistant sets in the unit hypercube

Ideal matrices and clutters are prevalent in Combinatorial Optimization, ranging from balanced matrices, clutters of T-joins, to clutters of rooted arborescences. Most of the known examples of ideal clutters are combinatorial in nature. In this paper, rendered by the recently developed theory of cuboids, we provide a different class of ideal clutters, one that is geometric in nature. The advantage of this new class of ideal clutters is that it allows for infinitely many ideal minimally non-packing clutters. We characterize the densest ideal minimally non-packing clutters of the class. Using the tools developed, we then verify the Replication Conjecture for the class

Scheherazade At Ground Zero: Muslim Women’s Agency, Identity, And Space In Euro-America From The 1893 World’s Columbian Exposition To The Islamic State

This dissertation examines three periods in Euro-American history that appear disparate but reflect the West’s changing relationship with the Islamic World and with Muslims, in particular with Muslim women, in Muslim-majority countries, the United States, and the United Kingdom. Specifically, this dissertation examines how Muslim women’s agency in the United States evolved from the erotic, provocative performance of the hootchy-kootchy at the 1893 World’s Columbian Exposition to the political and economic activism after post-September 11th, the Trump Era, and finally a different kind of agency through radicalization through the Islamic State (IS). The structure of this dissertation uses the voice of the narrator Scheherazade, from the 1001 Arabian Nights, co-opted by the Western literary canon, secularized, and made into a fairy tale princess through popular culture into a secular entity. It also uses the post-9/11 site of “Ground Zero” as both the actual and metaphorical site where Muslim women’s agency and visibility, dictated until this point by popular culture, media, and fairy tale, began to evolve into a more assertive presence both within the Muslim communities in the West but also as citizens of Western nations. The research for this dissertation draws upon a cross-section of the material culture from the Maghreb, the newly industrialized United States, political protest, entrepreneurship, and social media woven together to illustrate a century of navigating a religious, social, and political identity and citizenship borne of conflict with, and assimilation within, Western nations