Incidence Vectors Research Articles

The effect of COVID-19 on cancer incidences in the U.S

Fundamental data analysis assists in the evaluation of critical questions to discern essential facts and elicit formerly invisible evidence. In this article, we provide clarity into a subtle phenomenon observed in cancer incidences throughout the time of the COVID-19 pandemic. We analyzed the cancer incidence data from the American Cancer Society [1]. We partitioned the data into three groups: the pre-COVID-19 years (2017, 2018), during the COVID-19 years (2019, 2020, 2021), and the post-COVID-19 years (2022, 2023). In a novel manner, we applied principal components analysis (PCA), computed the angles between the cancer incidence vectors, and then added lognormal probability concepts in our analysis. Our analytic results revealed that the cancer incidences shifted within each era (pre, during, and post), with a meaningful change in the cancer incidences occurring in 2020, the peak of the COVID-19 era. We defined, computed, and interpreted the exceedance probability for a cancer type to have 1000 incidences in a future year among the breast, cervical, colorectal, uterine corpus, leukemia, lung & bronchus, melanoma, Hodgkin's lymphoma, prostate, and urinary cancers. We also defined, estimated, and illustrated indices for other cancer diagnoses from the vantage point of breast cancer in pre, during, and post-COVID-19 eras. The angle vectors post the COVID-19 were 72% less than pre-pandemic and 28% less than during the pandemic. The movement of cancer vectors was dynamic between these eras, and movement greatly differed by type of cancer. A trend chart of cervical cancer showed statistical anomalies in the years 2019 and 2021. Based on our findings, a few future research directions are pointed out.

Steiner Cut Dominants

For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut [Formula: see text] with [Formula: see text] and [Formula: see text]. The T-Steiner cut dominant of G is the dominant [Formula: see text] of the convex hull of the incidence vectors of the T-Steiner cuts of G. For [Formula: see text], this is the well-understood s-t-cut dominant. Choosing T as the set of all nodes of G, we obtain the cut dominant for which an outer description in the space of the original variables is still not known. We prove that for each integer τ, there is a finite set of inequalities such that for every pair (G, T) with [Formula: see text], the nontrivial facet-defining inequalities of [Formula: see text] are the inequalities that can be obtained via iterated applications of two simple operations, starting from that set. In particular, the absolute values of the coefficients and of the right-hand sides in a description of [Formula: see text] by integral inequalities can be bounded from above by a function of [Formula: see text]. For all [Formula: see text], we provide descriptions of [Formula: see text] by facet-defining inequalities, extending the known descriptions of s-t-cut dominants.

On the bond polytope

Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems

Let V be a set of n vertices, {mathcal M} a set of m labels, and let {textbf{R}} be an m times n matrix ofs independent Bernoulli random variables with probability of success p; columns of {textbf{R}} are incidence vectors of label sets assigned to vertices. A random instance G(V, E, {textbf{R}}^T {textbf{R}}) of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely [{textbf{R}}^T {textbf{R}}]_{v,u}) between any two vertices u, v for which this weight is strictly larger than 0. In this paper we study the average case analysis of Weighted Max Cut, assuming the input is a weighted random intersection graph, i.e. given G(V, E, {textbf{R}}^T {textbf{R}}) we wish to find a partition of V into two sets so that the total weight of the edges having exactly one endpoint in each set is maximized. In particular, we initially prove that the weight of a maximum cut of G(V, E, {textbf{R}}^T {textbf{R}}) is concentrated around its expected value, and then show that, when the number of labels is much smaller than the number of vertices (in particular, m=n^{alpha }, alpha <1), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case n=m and constant average degree (i.e. p = frac{Theta (1)}{n}), we show that with high probability, a majority type randomized algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we formally prove a connection between the computational problem of finding a (weighted) maximum cut in G(V, E, {textbf{R}}^T {textbf{R}}) and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system Sigma with incidence matrix {textbf{R}} (i.e. minimum imbalance over all sets in Sigma ). We exploit this connection by proposing a (weak) bipartization algorithm for the case m=n, p = frac{Theta (1)}{n} that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in a set system with incidence matrix {textbf{R}}. In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in G(V, E, {textbf{R}}^T {textbf{R}}). Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when p = frac{c}{n}, c<1.

Characterization of graphs with perfect closed neighbourhood matrices

The main purpose of this work is to characterize the graphs whose closed neighbourhood matrices are perfect since this property implies the resolution of some packing problems in polynomial time. Given a 0-1 matrix M, Q(M) denotes the graph whose cliques have their incidence vectors as the rows of M. A 0-1 matrix M is perfect if it is the clique-node matrix of Q(M) and Q(M) is a perfect graph.First, we define a set T of seven graphs and prove that N[G] is a clique-node matrix if and only if every node induced subgraph of G in the set T, has a common neighbour in G. Second, we define two families of graphs, called H-graphs and A-graphs, and prove that Q(N[G]) does not have an induced odd hole if and only if G has no H-subgraph for some additional conditions on G. Finally, we show that Q(N[G]) does not have an induced odd antihole if and only if G has neither an A-subgraph nor a web graph Wt4t+3 in suitable sets of nodes of V(G), thus completing a characterization of graphs with perfect closed neighbourhood matrix.

A Facet Defining of the Dicycle Polytope

In this paper, we consider the polytope <img src=image/13429471_01.gif> of all elementary dicycles of a digraph <img src=image/13429471_02.gif>. Dicycles problem, in graph theory and combinatorial optimization, solved by polyhedral approaches has been extensively studied in literature. Therefore cutting plane and branch and cut algorithms are unavoidable to exactly solve such a combinatorial optimization problem. For this purpose, we introduce a new family of valid inequalities called <img src=image/13429471_03.gif> alternating 3-arc path inequalities for the polytope of elementary dicycles <img src=image/13429471_01.gif>. Indeed, these inequalities can be used in cutting plane and branch and cut algorithms to construct strengthened relaxations of a linear formulation of the dicycle problem. To prove the facetness of <img src=image/13429471_03.gif> alternating 3-arc path inequalities, in opposite to what is usually done that consists basically to determine the affine subspace of a linear description of the considered polytope, we resort to constructive algorithms. Given the set of arcs of the digraph <img src=image/13429471_02.gif>, algorithms devised and introduced are based on the fact that from a first elementary dicycle, all other dicycles are iteratively generated by replacing some arcs of previously generated dicycles by others such that the current elementary dicycle contains an arc that does not belong to any other previously generated dicycles. These algorithms generate dicyles with affinely independent incidence vectors that satisfy <img src=image/13429471_03.gif> alternating 3-arc path inequalities with equality. It can easily be verified that all these devised algorithms are polynomial from time complexity point of view.

Finding all minimum cost flows and a faster algorithm for the K best flow problem

This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O ( F ( m + n ) + m n + M ) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher’s well-known method for the determination of the K best integer flows (Hamacher, 1995) at the first sub-optimal flow results in an algorithm with a running time of O ( F m ( n log n + m ) + M ) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so-called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O ( K n 3 + M ) . Besides, we give lower and upper bounds for the number of all optimal integer and feasible integer solutions. Our bounds are based on the fact that any optimal solution can be obtained by an initial optimal tree solution plus a conical combination of incidence vectors of all induced cycles with bounded coefficients.

A polyhedral view to a generalization of multiple domination

Given an undirected simple graph G=(V,E) and integer values fv,v∈V, a node subset D⊆V is called an f-tuple dominating set if, for each node v∈V, its closed neighborhood intersects D in at least fv nodes. We study the polytope that is defined as the convex hull of the incidence vectors in RV of the f-tuple dominating sets in G. New families of valid inequalities are introduced and a complete formulation is given for the case of stars. A corollary of our results is a proof that the conjecture reported in Argiroffo (2013) on a complete formulation of the 2-tuple dominating set polytope of trees does not hold. Preliminary computational results are also reported.

Box-total dual integrality and edge-connectivity

Given a graph $$G=(V,E)$$ and an integer $$k\ge 1$$ , the graph $$H=(V,F)$$ , where F is a family of elements (with repetitions allowed) of E, is a k-edge-connected spanning subgraph of G if H cannot be disconnected by deleting any $$k-1$$ elements of F. The convex hull of incidence vectors of the k-edge-connected subgraphs of a graph G forms the k-edge-connected subgraph polyhedron of G. We prove that this polyhedron is box-totally dual integral if and only if G is series–parallel. In this case, we also provide an integer box-totally dual integral system describing this polyhedron.

The biclique partitioning polytope

Given a bipartite graph G=(V1,V2,E), a biclique of G is a complete bipartite subgraph of G, and a biclique partitioning of G is a set A⊆E such that the bipartite graph G′=(V1,V2,A) is a vertex-disjoint union of bicliques. The biclique partitioning problem (BPP) consists of, given a complete bipartite graph G=(V1,V2,E) with edge weights we∈R for all e∈E (thus, negative weights are allowed), finding a biclique partitioning A⊆E of minimum weight. The bicluster editing problem (BEP) is a variant of the BPP and consists of editing a minimum number of edges of an input bipartite graph G in order to transform its edge set into a biclique partitioning. Editing an edge consists of either adding it to the graph or deleting it from the graph. In addition to the BPP and the BEP, other problems in the literature aim at finding biclique partitionings, and this motivates the study of the biclique partitioning polytope Pnm of the complete bipartite graph Knm (i.e., Pnm is the convex hull of the incidence vectors of the biclique partitionings of Knm). In this paper we develop such a polyhedral study and show that ladder, bellows, and grid inequalities induce facets of Pnm. Our computational results show that these inequalities are very effective in solving the BEP. In particular, they are able to improve the value of the relaxed solution by up to 20%.

Polyhedral properties of the induced cluster subgraphs

A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph G, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope associated with G is defined as the convex hull of the incidence vectors of all IUCs in the graph. The Maximum IUC problem, which is to find a maximum-cardinality IUC in a graph, finds applications in network-based data analysis. In this paper, we derive several families of facet-defining valid inequalities for the IUC polytope. We also give a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problem for most of the considered families of valid inequalities and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the Maximum IUC problem through computational experiments.

Small weight code words arising from the incidence of points and hyperplanes in $$\text {PG}(n,q)$$

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.

Birkhoff--von Neumann Graphs that are PM-Compact

A well-studied geometric object in combinatorial optimization is the perfect matching polytope of a graph $G$---the convex hull of the incidence vectors of all perfect matchings of $G$. In any investigation concerning the perfect matching polytope, one may assume that $G$ is matching covered---that is, $G$ is a connected graph (of order at least two) and each edge of $G$ lies in some perfect matching. A graph $G$ is Birkhoff--von Neumann if its perfect matching polytope is characterized solely by nonnegativity and degree constraints. A result of Balas [North Holland Math. Stud., 59 (1981), pp. 1--13] implies that $G$ is Birkhoff--von Neumann if and only if $G$ does not contain a pair of vertex-disjoint odd cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. It follows immediately that the corresponding decision problem is in co-$\mathcal{NP}$. However, it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is planar---due to a result of Carvalho, Lucchesi, and Murty [J. Combin. Theory Ser. B, 92 (2004), pp. 319--324]. More recently, these authors, along with Kothari [SIAM J. Discrete Math., 32 (2018), pp. 1478--1504], have shown that this problem is equivalent to the seemingly unrelated problem of deciding whether a given graph is $\overline{C_6}$-free. The combinatorial diameter of a polytope is the diameter of its $1$-skeleton graph. A graph $G$ is PM-compact if the combinatorial diameter of its perfect matching polytope equals one. Independent results of Balinski and Russakoff [SIAM Rev., 16 (1974), pp. 516--525] and of Chvátal [J. Combin. Theory Ser. B, 18 (1975), pp. 138--154] imply that $G$ is PM-compact if and only if $G$ does not contain a pair of vertex-disjoint even cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. Once again the corresponding decision problem is in co-$\mathcal{NP}$, but it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is bipartite or is near-bipartite---due to a result of Wang et al. [Discrete Math., 313 (2013), pp. 772--783]. In this paper, we consider the “intersection” of the aforementioned problems. We give an alternative description of matching covered graphs that are Birkhoff--von Neumann as well as PM-compact; our description implies that the corresponding decision problem is in $\mathcal{P}$.

Valid Inequalities for the Maximal Matching Polytope

Given a graph G=(V,E), a subset M of E is called a matching if no two edges in M are adjacent. A matching is said to be maximal if it is not a proper subset of any other matching. The maximal matching polytope associated with graph G is the convex hull of the incidence vectors of maximal matchings in G . In this paper, we introduce new classes of valid inequalities for the maximal matching polytope.

On the star forest polytope for trees and cycles

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

Shorter tours and longer detours: uniform covers and a bit beyond

Motivated by the well known “four-thirds conjecture” for the traveling salesman problem (TSP), we study the problem of uniform covers. A graph $$G=(V,E)$$ has an $$\alpha $$ -uniform cover for TSP (2EC, respectively) if the everywhere $$\alpha $$ vector (i.e., $$\{\alpha \}^{E}$$ ) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides’ algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Sebő asked if such graphs have $$(1-\epsilon )$$ -uniform covers for TSP for some $$\epsilon > 0$$ . Indeed, the four-thirds conjecture implies that such graphs have $$\frac{8}{9}$$ -uniform covers. We show that these graphs have $$\frac{18}{19}$$ -uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere $$\frac{15}{17}$$ vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere $$\frac{2}{3}$$ vector is an optimal solution for the subtour elimination linear programming relaxation for TSP, then a tour with weight at most $$\frac{27}{19}$$ times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. An essential ingredient in our proofs is decompositions of graphs (e.g., cycle covers) that cover small-cardinality cuts an even (nonzero) number of times. Another essential tool we use is half-integral tree augmentation, which is known to have a small integrality gap. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose a point in the subtour elimination polytope into spanning, connected subgraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a $$\frac{17}{12}$$ -approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs.

Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.

Skeleton matching polytope: Realization and isomorphism

Let G be a simple graph. A set of pairwise disjoint edges is a matching of G. The matching polytope of G, denoted M(G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G, SMP(G) or, even simpler, SMP. In this paper, we consider the following questions: (1) Which graphs realize SMPs? (2) Given two non-isomorphic graphs, are their skeletons non-isomorphic as well? Concerning the first question, it is well-known (see Naddef and Pulleyblank (1984) and Balinski (1961) [3]) that neither non-connected graphs nor non-Hamiltonian graphs realize SMPs. Among other simple results, we characterize both planar graphs and bipartite graphs that occur as SMPs. The second question is completely solved here. For every G and H, other than G=G′∪i=1kGi and H=H′∪i=1kHi where, for i=1,…,k, Gi and Hi are isomorphic to K3 or S1,3 and G′≃H′, we prove that their skeleton matching polytopes, G(M(G)) and G(M(H)), are isomorphic graphs if and only if G and H are also isomorphic.

A Note on the Matching Polytope of a Graph

The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of G(M(G)) is equal to the number of edges of G, generalizing a known result for trees. From this, we identify the vertices of the skeleton with the minimum degree and we prove that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.

On f-domination: polyhedral and algorithmic results

Given an undirected simple graph G with node set V and edge set E, let $$f_v$$ , for each node $$v \in V$$ , denote a nonnegative integer value that is lower than or equal to the degree of v in G. An f-dominating set in G is a node subset D such that for each node $$v\in V{{\setminus }}D$$ at least $$f_v$$ of its neighbors belong to D. In this paper, we study the polyhedral structure of the polytope defined as the convex hull of all the incidence vectors of f-dominating sets in G and give a complete description for the case of trees. We prove that the corresponding separation problem can be solved in polynomial time. In addition, we present a linear-time algorithm to solve the weighted version of the problem on trees: Given a cost $$c_v\in {\mathbb {R}}$$ for each node $$v\in V$$ , find an f-dominating set D in G whose cost, given by $$\sum _{v\in D}{c_v}$$ , is a minimum.