Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2009 Exploring Agreeability in Tree Societies Sarah Fletcher Harvey Mudd College Recommended Citation Fletcher, Sarah, "Exploring Agreeability in Tree Societies" (2009). HMC Senior Theses. 218. https://scholarship.claremont.edu/hmc_theses/218 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Exploring Agreeability in Tree Societies Sarah Fletcher Francis Su, Advisor Kimberly Tucker, Reader May, 2009 Department of Mathematics Copyright c 2009 Sarah Fletcher. The author grants Harvey Mudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy- ing is by permission of the author. To disseminate otherwise or to republish re- quires written permission from the author. Abstract Let S be a collection of convex sets in Rd with the property that any sub- collection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that \s2S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one-dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection prop- erties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty inter- section. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that 1 the agreement number of (2, m)-agreeable tree societies is at least 3 jSj and that the agreement number of (k, k + 1)-agreeable tree societies is at least jSj − 1. Contents Abstract iii Acknowledgments xi 1 Background 1 1.1 Helly’s Theorem . 1 1.2 Agreeable Societies . 2 1.3 Graphs and the Helly Property . 5 1.4 Agreeability on Graphs . 5 1.5 Terminology and Notation . 6 1.6 Our Main Goal . 7 2 A Reduction Method 9 2.1 Equivalent Tree Societies . 9 2.2 Irreducible Tree Societies . 9 2.3 A Reduction Process . 11 3 (k, m)-Theorems for Tree Societies 15 3.1 Definitions . 15 3.2 (n, n + 1)-Agreeable Tree Societies . 16 3.3 (2, m)-Agreeable Tree Societies . 18 3.4 More Small Cases . 21 4 Future Work and Discussion 29 4.1 Comparing Linear Societies and Tree Societies . 29 4.2 Other Possible Directions . 30 Bibliography 31 List of Figures 1.1 Four pairwise intersecting intervals. 1 1.2 Three convex sets that intersect pairwise but have a non- empty intersection. 2 1.3 Interpreting the real line as a political spectrum. 3 1.4 A discrete linear society. 4 2.1 Two equivalent societies. 10 2.2 Kinds of leaves. 10 2.3 Illustration of reduction algorithm. 13 3.1 (2, 3)-agreeable tree society with minimal agreement number. 17 3.2 (2, 4)-agreeable tree society with minimal agreement number. 21 3.3 (3, 5)-agreeable tree society with minimal agreement number. 22 3.4 A (3, 6)-agreeable reduced tree society with four leaves and agreement number jSj − 4. 24 3.5 A (3, 6)-agreeable reduced tree society with three leaves and l jSj−1 m a(S) = 2 ........................... 26 List of Tables 3.1 Summary of Small Cases. 27 Acknowledgments I would like to thank Professor Francis Su for his patient guidance and en- couragement both as my advisor for this thesis project and throughout my four years at Mudd and my second reader Professor Kimberly Tucker for all of her helpful comments. I would also like to thank the math depart- ment as a whole for providing a wonderful and supportive environment to learn and grow in for the last four years. Chapter 1 Background 1.1 Helly’s Theorem The simplest version of the classical Helly’s theorem states: Helly’s Theorem in 1-dimension. Given n convex sets, that is intervals, in R, if every pair has a nonempty intersection then all n intersect at a common point. Consider the example in Figure 1.1. Each of the colored lines represents an interval of R, slightly displaced so they may be distinguished. Each pair of intervals has a nonempty intersection and we can see that there is a point contained in all of the intervals. On the plane, the property that every pair of sets in a collection of con- vex sets has a nonempty intersection is no longer sufficient to guarantee that there will be a point contained in all of the sets. In Figure 1.2, we see three convex sets that intersect pairwise but still have an empty intersec- Figure 1.1: Four pairwise intersecting intervals. 2 Background Figure 1.2: Three convex sets that intersect pairwise but have a nonempty intersection. tion. However, if we change the hypothesis to require that every three sets in a collection of n convex sets has a nonempty intersection, we guaran- tee that there is a point in the intersection of the n sets. In general, the d-dimensional version of the classical Helly’s theorem states: Theorem 1.1 (Helly). Given n convex sets in Rd where n > d, if every d + 1 of them has a nonempty intersection, then they all intersect at a common point. In moving to higher dimensions, it is necessary to change the intersec- tion properties of the sets in the hypothesis to guarantee the same conclu- sion. It is also interesting to consider what results can be obtained when we alter the intersection properties in the hypothesis. This has been well ex- plored by Berg et al. (Forthcoming) both in the linear case and in Rd. In the linear case, Berg et al. (Forthcoming) also provide an interesting application to approval voting systems, which we describe below. 1.2 Agreeable Societies If we consider R as the political spectrum and an interval (convex set) as the region of the political spectrum of which a particular voter approves , we can use a collection of intervals to model the political preferences of a society. For example, in Figure 1.3 we have taken our earlier linear example and labeled the line as a political spectrum. We can interpret the line and collection of four intervals as a society with four voters and see that there is a region of the political spectrum approved by all of the voters. Agreeable Societies 3 Liberal Conservative Figure 1.3: Interpreting the real line as a political spectrum. If X is a linear space and S is a collection of convex sets of X, we call the ordered pair (X, S) a linear society, view X as a political spectrum, and associate a voter v to each element Av 2 S called the voter’s approval set. If a candidate’s position falls within Av we say that the voter v would approve that candidate. Let the agreement number a(S) of a society (X, S) be the size of the largest subset of S that has a nonempty mutual intersection. In a society satisfying the hypotheses of the classical Helly’s Theorem, such as the society in Figure 1.3, we have a(S) = jSj. Translating Helly’s theorem in R using this context, we see that in a linear society (R, S) in which the convex sets corresponding to every pair of voters have a nonempty intersection, we have a(S) = jSj, that is, there is some point on the political spectrum of which all voters in the society approve. However, if we are having an election, it is not terribly useful to have a point on the spectrum approved by all voters if there is no candidate at that point for whom to vote. To incorporate the idea of candidates, we can either assume that there is a candidate at every point on the line or we can place a node for every candidate on the line thus moving from R into a discrete line, that is, a possibly infinite (if there are infinitely many candidates) linear tree. If we move to this discrete scenario, then we require the subsets corresponding to the voters to be subtrees of the underlying linear tree. In Figure 1.4 we add five candidates to the linear society we were looking at earlier and convert the intervals associated with each of the four voters into subtrees. Furthermore, we know that any given group of people is highly un- likely to all agree on anything, so it is interesting to ask what happens if, for example, we know that some pair amongst every three people in a so- ciety agree even though any given pair might not. The following theorem gives a bound on the agreement number of a linear society in which every 4 Background Figure 1.4: A discrete linear society. collection of m voters has some subset of k voters that all approve of some candidate.