Combinatorics and Optimization

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

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 12 of 12

Approximating Minimum-Size 2-Edge-Connected and 2-Vertex-Connected Spanning Subgraphs
(University of Waterloo, 2017-04-27) Narayan, Vishnu Verambudi; Cheriyan, Joseph
We study the unweighted 2-edge-connected and 2-vertex-connected spanning subgraph problems. A graph is 2-edge-connected if it is connected on removal of an edge, and it is 2-vertex-connected if it is connected on removal of a vertex. The problem of finding a minimum-size 2-edge-connected (or 2-vertex-connected) spanning subgraph of a given graph is NP-hard. We present a 4/3-approximation algorithm for unweighted 2ECSS on 3-vertex-connected input graphs, which matches the best known approximation ratio due to Sebő and Vygen for the general unweighted 2ECSS problem, but our analysis is with respect to the D2 lower bound. We also give a 17/12-approximation algorithm for unweighted 2VCSS on graphs of minimum degree at least 3, which is lower than the best known ratios of 3/2 by Garg, Santosh and Singla and 10/7 by Heeger and Vygen for the general unweighted 2VCSS problem. These algorithms are accompanied by new theorems about the known lower bounds.
Augmenting Trees to Achieve 2-Node-Connectivity
(University of Waterloo, 2020-09-02) Grout, Logan; Cheriyan, Joseph
This thesis focuses on the Node-Connectivity Tree Augmentation Problem (NC-TAP), formally defined as follows. The first input of the problem is a graph G which has vertex set V and edge set E. We require |V| >= 3 to avoid degenerate cases. The edge set E is a disjoint union of two sets T and L where the subgraph (V,T) is connected and acyclic. We call the edges in T the tree edges and the edges in L are called links. The second input is a vector c in R^L, c >= 0 (a vector of nonnegative real numbers indexed by the links), which is called the cost of the links. We often refer to this graph G and cost vector c as an instance of NC-TAP. Given an instance G = (V, T U L) and c to NC-TAP, a feasible solution to that instance is a set of links F such that the graph (V, T U F) is 2-connected. The cost of a set of links. The goal of NC-TAP is to find a feasible solution F^* to the given instance such that the the cost of F^* is minimum among all feasible solutions to the instance. This thesis is mainly expository and it has two goals. First, we present the current best-known algorithms for NC-TAP. The second goal of this thesis is to explore new directions in the study of NC-TAP in the last chapter. This is an exploratory chapter where the goal is to use the state of the art techniques for TAP to develop an algorithm for NC-TAP which has an approximation guarantee better than factor 2.
Brick Generation and Conformal Subgraphs
(University of Waterloo, 2016-04-15) Kothari, Nishad; Cheriyan, Joseph; Murty, U. S. R.
A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, called bricks (which are nonbipartite and 3-connected) and braces (which are bipartite). A fundamental theorem of LovU+00E1sz (1983) states that every nonbipartite matching covered graph admits an ear decomposition starting with a bi-subdivision of $K_4$ or of the triangular prism $\overline{C_6}$. This led Carvalho, Lucchesi and Murty (2003) to pose two problems: (i) characterize those nonbipartite matching covered graphs which admit an ear decomposition starting with a bi-subdivision of $K_4$, and likewise, (ii) characterize those which admit an ear decomposition starting with a bi-subdivision of $\overline{C_6}$. In the first part of this thesis, we solve these problems for the special case of planar graphs. In Chapter 2, we reduce these problems to the case of bricks, and in Chapter 3, we solve both problems when the graph under consideration is a planar brick. A nonbipartite matching covered graph G is near-bipartite if it has a pair of edges U+03B1 and U+03B2 such that G-{U+03B1,U+03B2} is bipartite and matching covered; examples are $K_4$ and $\overline{C_6}$. The first nonbipartite graph in any ear decomposition of a nonbipartite graph is a bi-subdivision of a near-bipartite graph. For this reason, near-bipartite graphs play a central role in the theory of matching covered graphs. In the second part of this thesis, we establish generation theorems which are specific to near-bipartite bricks. Deleting an edge e from a brick G results in a graph with zero, one or two vertices of degree two, as G is 3-connected. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of G-e is the graph J obtained from it by bicontracting all its vertices of degree two. The edge e is thin if J is also a brick. Carvalho, Lucchesi and Murty (2006) showed that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a thin edge. In general, given a near-bipartite brick G and a thin edge e, the retract J of G-e need not be near-bipartite. In Chapter 5, we show that every near-bipartite brick G, distinct from $K_4$ and $\overline{C_6}$, has a thin edge e such that the retract J of G-e is also near-bipartite. Our theorem is a refinement of the result of Carvalho, Lucchesi and Murty which is appropriate for the restricted class of near-bipartite bricks. For a simple brick G and a thin edge e, the retract of G-e may not be simple. It was established by Norine and Thomas (2007) that each simple brick, which is not in any of five well-defined infinite families of graphs, and is not isomorphic to the Petersen graph, has a thin edge such that the retract J of G-e is also simple. In Chapter 6, using our result from Chapter 5, we show that every simple near-bipartite brick G has a thin edge e such that the retract J of G-e is also simple and near-bipartite, unless G belongs to any of eight infinite families of graphs. This is a refinement of the theorem of Norine and Thomas which is appropriate for the restricted class of near-bipartite bricks.
Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs
(University of Waterloo, 2022-08-17) Qu, Zishen; Cheriyan, Joseph; Liu, Yu-Ru
A sieve method is in effect an application of the inclusion-exclusion counting principle, and the estimation methods to avoid computing the explicit formula. Sieve methods have been used in number theory for over a hundred years. These methods have been modified to make use of the structure of integer-like objects; producing better estimates and providing more use cases. The first part of the thesis aims to analyze and use the analogues of number theoretic sieves in combinatorial contexts. This part consists of my work with Yu-Ru Liu in Chapters 2 and 3. We focus on two sieve methods: the Turán sieve (introduced by Liu and Murty in 2005) and the Selberg sieve (independently generalized by Wilson in 1969 and Chow in 1998 with slightly different formulations). Some comparisons and applications of these sieves are discussed. In particular, we apply the combinatorial Turán sieve to count labelled graphs and we apply the combinatorial Selberg sieve to count subspaces of finite spaces. Finding sufficient conditions for Hamiltonicity in graphs is a classical topic, where the difficulty is bracketed by the NP-hardness of the associated decision problem. The second part of the thesis, consisting of Chapter 4, aims to characterize Hamiltonicity by means of induced subgraphs. The results in this chapter are based on the paper "Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs." Discrete Mathematics, 345(7):112869, 2022, co-authored with Joseph Cheriyan, Sepehr Hajebi, and Sophie Spirkl. We study induced subgraphs and conditions for Hamiltonicity. In particular, we characterize the minimal 2-connected non-Hamiltonian split graphs and the minimal 2-connected non-Hamiltonian triangle-free graphs.
Combinatorially Thin Trees and Spectrally Thin Trees in Structured Graphs
(University of Waterloo, 2023-12-19) Alghasi, Mahtab; Cheriyan, Joseph; Tunçel, Levent
Given a graph $G=(V,E)$, finding simpler estimates of $G$ with possibly fewer edges or vertices while capturing some of its specific properties has been used in order to design efficient algorithms. The concept of estimating a graph with a simpler graph is known as graph sparsification. Spanning trees are an important family of graph sparsifiers that maintain connectivity of graphs, and have been utilized in many applications. However, spanning trees are a wide family, and for some applications one might need the spanning tree to have specific properties. Combinatorially thin trees are a type of spanning trees that show up in applications such as Asymmetric Travelling Salesman Problem (ATSP). A spanning tree $T$ of $G$ is combinatorially thin if there is no cut $U\subset V$ such that $T$ contains all the edges in $\delta(U)$, and the thinness parameter $\alpha_G(T)$ measures the maximum fraction of edges in $E(T)\cap \delta(U)$ compared to $\delta(U)$ over all cuts $U\subset V$. Intuitively, combinatorial thinness measures how much edge-connectivity we lose while removing the spanning tree $T$ from $G$. It is easy to verify that if $G$ has connectivity $k$, then $\frac{1}{k}$ lower bounds $\alpha_G$. On the other hand, Goddyn conjectured that $\alpha_G$ can also be upper bounded as a function of connectivity $\alpha_G = f(\frac{1}{k})$. This conjecture which is known as thin tree conjecture, was proved for the special case of graphs with bounded genus by Oveis-Gharan and Saberi, in 2011. However, the general case is still open. In the first part of this thesis, we study some of the known connections between edge-connectivity and $\alpha_{G}$ and investigate the result of Oveis-Gharan and Saberi for the special case of planar graphs. For a general graph $G$ and spanning tree $T$, even verifying the combinatorial thinness $\alpha_{G}(T)$ of $T$ is an $\text{NP}$-hard problem. A natural more efficiently computable relaxation of combinatorial thinness is the notion of spectral thinness. For a graph $G$ and a spanning tree $T$ in $G$ the spectral thinness $\theta_{G}(T)$ is the smallest value of $\theta$ such that $\theta\L_G - \L_T$ is a positive semidefinite matrix where $\L_G$ and $\L_T$ are Laplacian matrices of $G$ and $T$. Additionally, we define $\theta_G$ to be the minimum value of $\theta_{G}(T)$ over all spanning trees $T$ of $G$. Similar to combinatorial thinness and connectivity, $\theta_{G}(T)$ can be lower bounded by the maximum effective resistance of edges in $T$. It was also proven by Harvey and Olver in 2014 that the maximum effective resistance of edges in $G$ asymptotically upper bounds $\theta_{G}$. However, finding a mathematical characterization of $\theta_{G}(T)$, even for structured graphs, is still a challenge. In the second part of this thesis, we will give general lower bound and upper bound certificates for $\theta_{G}(T)$ and utilize these certificates for circulant matrices to estimate spectral thinness of graphs such as complete graphs, complete bipartite graphs, and prism graphs.
Decomposition-based methods for Connectivity Augmentation Problems
(University of Waterloo, 2021-09-03) Neogi, Rian; Cheriyan, Joseph
In this thesis, we study approximation algorithms for Connectivity Augmentation and related problems. In the Connectivity Augmentation problem, one is given a base graph G=(V,E) that is k-edge-connected, and an additional set of edges $L \subseteq V\times V$ that we refer to as links. The task is to find a minimum cost subset of links $F \subseteq L$ such that adding F to G makes the graph (k+1)-edge-connected. We first study a special case when k=1, which is equivalent to the Tree Augmentation problem. We present a breakthrough result by Adjiashvili that gives an approximation algorithm for Tree Augmentation with approximation guarantee below 2, under the assumption that the cost of every link $\ell \in L$ is bounded by a constant. The algorithm is based on an elegant decomposition based method and uses a novel linear programming relaxation called the $\gamma $-bundle LP. We then present a subsequent result by Fiorini, Gross, Konemann and Sanita who give a $3/2+\epsilon$ approximation algorithm for the same problem. This result uses what are known as Chvatal-Gomory cuts to strengthen the linear programming relaxation used by Adjiashvili, and uses results from the theory of binet matrices to give an improved algorithm that is able to attain a significantly better approximation ratio. Next, we look at the special case when k=2. This case is equivalent to what is known as the Cactus Augmentation problem. A recent result by Cecchetto, Traub and Zenklusen give a 1.393-approximation algorithm for this problem using the same decomposition based algorithmic framework given by Adjiashvili. We present a slightly weaker result that uses the same ideas and obtains a $3/2+\epsilon $ approximation ratio for the Cactus Augmentation problem. Next, we take a look at the integrality ratio of the natural linear programming relaxation for Tree Augmentation, and present a result by Nutov that bounds this integrality gap by 28/15. Finally, we study the related Forest Augmentation problem that is a generalization of Tree Augmentation. There is no approximation algorithm for Forest Augmentation known that obtains an approximation ratio below 2. We show that we can obtain a 29/15-approximation algorithm for Forest Augmentation under the assumption that the LP solution is half-integral via a reduction to Tree Augmentation. We also study the structure of extreme points of the natural linear programming relaxation for Forest Augmentation and prove several properties that these extreme points satisfy.
Dynamic Pricing Schemes in Combinatorial Markets
(University of Waterloo, 2023-05-01) Kalichman, David; Cheriyan, Joseph; Pashkovich, Kanstantsin
In combinatorial markets where buyers are self-interested, the buyers may make purchases that lead to suboptimal item allocations. As a central coordinator, our goal is to impose prices on the items of the market so that its buyers are incentivized to exclusively make optimal purchases. In this thesis, we study the question of whether dynamic pricing schemes can achieve the optimal social welfare in multi-demand combinatorial markets. This well-motivated question has been the topic of some study, but has remained mostly open, and to date, positive results are only known for extremal cases. In this thesis, we present the current results for unit-demand, bi-demand and tri-demand markets. In the context of these results, we discuss the significance of not having a deficiency of items, which is known as the (OPT) condition. We outline an approach for handling an item deficiency, and we expose barriers to extending the known techniques to markets of larger demand.
Extensions of Galvin's Theorem
(University of Waterloo, 2018-04-30) Levit, Maxwell; Cheriyan, Joseph
We discuss problems in list coloring with an emphasis on techniques that utilize oriented graphs. Our central theme is Galvin's resolution of the Dinitz problem (Galvin. J. Comb. Theory, Ser. B 63(1), 1995, 153--158). We survey the related work of Alon and Tarsi (Combinatorica 12(2) 1992, 125--134) and H\"{a}ggkvist and Janssen (Combinatorics, Probability \& Computing 6(3) 1997, 295--313). We then prove two new extensions of Galvin's theorem.
The Matching Augmentation Problem: A 7/4-Approximation Algorithm
(University of Waterloo, 2019-05-23) Dippel, Jack; Cheriyan, Joseph
We present a 7/4 approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. We first present a series of approximation guarantee preserving reductions, each of which can be performed in polytime. Performing these reductions gives us a restricted collection of MAP instances. We present a 7/4 approximation algorithm for this restricted set of MAP instances. The algorithm starts with a subgraph which is a min-cost 2-edge cover, contracts its blocks, adds paths to the subgraph to cover all its bridges, and finally adds cycles to the subgraph to connect all its components. We contract any blocks created throughout. The algorithm ends when the subgraph is a single vertex, and we output all the edges we’ve contracted which form a 2ECSS.
Matchings and Representation Theory
(University of Waterloo, 2018-12-20) Lindzey, Nathan; Cheriyan, Joseph; Godsil, Chris
In this thesis we investigate the algebraic properties of matchings via representation theory. We identify three scenarios in different areas of combinatorial mathematics where the algebraic structure of matchings gives keen insight into the combinatorial problem at hand. In particular, we prove tight conditional lower bounds on the computational complexity of counting Hamiltonian cycles, resolve an asymptotic version of a conjecture of Godsil and Meagher in Erdos-Ko-Rado combinatorics, and shed light on the algebraic structure of symmetric semidefinite relaxations of the perfect matching problem
Thin Trees in Some Families of Graphs
(University of Waterloo, 2018-04-25) Mousavi Haji, Seyyed Ramin; Cheriyan, Joseph
Let 𝐺=(𝑉,𝐸) be a graph and let 𝑇 be a spanning tree of 𝐺. The thinness parameter of 𝑇 denoted by 𝜌(𝑇) is the maximum over all cuts of the proportion of the edges of 𝑇 in the cut. Thin trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP). Goddyn conjectured that every graph of sufficiently large edge-connectivity has a spanning tree 𝑇 such that 𝜌(𝑇) ≤ 𝜀. In this thesis, we study the problem of finding thin spanning trees in two families of graphs, namely, (1) distance-regular graphs (DRGs), and (2) planar graphs. For some families of DRGs such as strongly regular graphs, Johnson graphs, Crown graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees 𝑇 of maximum degree ≤ 3 such that 𝜌(𝑇) is determined by the parameters of the graph. For planar graphs, we improve the analysis of Merker and Postle ("Bounded Diameter Arboricity", arXiv:1608.05352v1) and show that every 6-edge-connected planar graph has two edge-disjoint spanning trees 𝑇,𝑇′ such that 𝜌(𝑇),𝜌(𝑇′) ≤ 14⁄15. For 8-edge-connected planar graphs 𝐺, we present a simplified version of the techniques of Merker and Postle and show that 𝐺 has two edge-disjoint spanning trees 𝑇,𝑇′ such that 𝜌(𝑇),𝜌(𝑇′) ≤ 12⁄13.
The Traveling Tournament Problem
(University of Waterloo, 2022-08-17) Bendayan, Salomon; Cheriyan, Joseph
In this thesis we study the Traveling Tournament problem (TTP) which asks to generate a feasible schedule for a sports league such that the total travel distance incurred by all teams throughout the season is minimized. Throughout our three technical chapters a wide range of topics connected to the TTP are explored. We begin by considering the computational complexity of the problem. Despite existing results on the NP-hardness of TTP, the question of whether or not TTP is also APX-hard was an unexplored area in the literature. We prove the affirmative by constructing an L-reduction from (1,2)-TSP to TTP. To reach the desired result, we show that given an instance of TSP with a solution of cost K, we can construct an instance of TTP with a solution of cost at most 20m(m+1)cK where m = c(n-1)+1, n is the number of teams, and c > 5, c ∈ ℤ is fixed. On the other hand, we show that given a feasible schedule to the constructed TTP instance, we can recover a tour on the original TSP instance. The next chapter delves into a popular variation of the problem, the mirrored TTP, which has the added stipulation that the first and second half of the schedule have the same order of match-ups. Building upon previous techniques, we present an approximation algorithm for constructing a mirrored double round-robin schedule under the constraint that the number of consecutive home or away games is at most two. We achieve an approximation ratio on the order of 3/2 + O(1)/n. Lastly, we present a survey of local search methods for solving TTP and discuss the performance of these techniques on benchmark instances.

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