An Approximation Algorithm for the Stable Marriagae Problem with Ties and Incomplete Lists

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations An Approximation Algorithm for the Stable Marriagae Problem with Ties and Incomplete Lists Rommel Jalasutram Clemson University Follow this and additional works at: Part of the Computer Sciences Commons Recommended Citation Jalasutram, Rommel, "An Approximation Algorithm for the Stable Marriagae Problem with Ties and Incomplete Lists" (2014). All Dissertations This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact

2 An Approximation Algorithm for the Stable Marriage Problem with Ties and Incomplete Lists A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Computer Science by Rommel Jalasutram December 2014 Accepted by: Dr. Brian C. Dean, Committee Chair Dr. Amy Apon Dr. Ilya Safro Dr. Pradip K. Srimani

3 Abstract Consider the bipartite matching problem with two sets of participants: men (L) and women (R). Each person participating has a strict and complete preference list over participants from the other set. The goal is to pair men and women such that no one can improve their partner by breaking away from the centralized matching scheme. Problems that exhibit this flavor are commonly classified as ordinal matchings. Gale and Shapley showed how to obtain such a matching (otherwise known as a stable matching). Generalizations of this model have found relevance in several centralized matching schemes such as the National Residency Matching Program where graduating doctors are matched to hospitals, human organ transplant exchange markets, and housing allocation markets. In this thesis, we study a generalization of the stable matching problem, where the preference lists of participants are allowed to contain ties and missing entries. Here, we seek to find a stable matching of maximum cardinality. The problem was shown to be NP-hard, and is one of the most prominent problems in the domain of ordinal matching. When preference lists are restricted to one side of the problem, Iwama et al. [28] devised a variant of the famous Gale-Shapley stable matching algorithm that breaks ties using edge weights obtained by solving a linear programming relaxation of the problem, leading to an approximation ratio of 25/17 (approximately ). We apply ideas from factor-revealing LPs to show that this analysis in [28] is an upper bound, but that the algorithm and analysis can be improved to yield an approximation ratio of at most 19/13 (approximately ), improving the best currently-known approximation ratio (obtained via different techniques in [39]) of 41/28 (approximately ). ii

4 Dedicated to my father, (late) Dr. Jalasutram Muralidhar iii

5 Acknowledgments My stay in graduate school has been longer than I had imagined. This very day seven years ago, I left my country to pursue higher studies. It has turned out to be a rather interesting journey, and many people I have come to know are responsible for it. First, I would like to thank my advisor Dr. Brian Dean for accepting me as his student. I consider myself to have been extremely lucky to have him as my advisor. Dr. Dean has encouraged me at every step, was extremely patient with me and was generous with his time, ideas, and gave me the freedom to choose my topic of interest. Without his optimism, I would still be stuck in a cave trying to find my way out. His motto to keep things simple: Mother Nature never intended it to be this complicated, has helped in simplifying the results presented in this dissertation. Working with Dr. Dean has been a great pleasure and an honor. I would like to thank Dr. Amy Apon, Dr. Ilya Safro, and Dr. Pradip Srimani for being on my committee. Next, I am grateful to Dexter Stowers and Dr. Wayne Madison for helping me secure funding during my first year of PhD studies. Without their help, I could not have started this journey. Working in the Applied Algorithms Lab has been a fun filled experience. I would like to thank my group members Raghuveer Mohan, Chad Waters and Matt Dabney for making sure I had a great time throughout. Over the years, I have made really amazing friends who have stood by me throughout and made my stay at Clemson memorable. Of the top of my head, I wish to acknowledge Achal Singhal, Biswa Singh, Sandeep Lokala, Kalaivani Sundararajan, Sumod Mohan, Dhananjay Joshi, Kirti Kanitkar, Shivkumar Morkhande, Sahasranshu Panda, Keya Sharma, Biswajit Mazumdar, Uttara Thakre, Rachana Ranade and many others for being there. Finally, it goes without saying that I owe everything I have achieved to my parents, (late) Dr. Jalasutram Muralidhar and Jalasutram Durga, and my younger brother Kartik. They made sure I had a happy childhood which continues to this day. My parents-in-law, Dr. Srikanth Guruswamy iv

6 and Kanakadhara Srikanth, and my sister-in-law Abhinaya have given me a second family in the last few years. They have been equally warm and loving as my own family. Last but not the least, my wife, Ahalya, has been my pillar of strength, who put up with my ups and downs and made sure I was well taken care of. This research has been supported in part by Dr. Dean s NSF career award CCF v

7 Table of Contents Title Page i Abstract ii Dedication iii Acknowledgments iv List of Tables vii List of Figures viii 1 Introduction Stable Matchings Stable Matching Generalizations Main Contributions Background Preliminaries Symmetric Difference and Augmenting Paths Linear Programming Relaxation Stable Allocation Relaxation Counterexamples and Lessons Learned Iterative and Randomized Rounding Deterministic Primal LP Rounding Stable Allocation Variants Factor-Revealing LP and Approximation GS-LP Algorithm Longer Augmenting Paths and Validity Towards a Factor-Revealing LP Conclusions and Discussion Bibliography vi

8 List of Tables 5.1 Different types of R edges. These edges come into existence due to GS-LP. Columns 2 and 3 represent the corresponding preference lists of man m 1 and woman w Approximation Ratios for LP f (d) vii

9 List of Figures 1.1 A maximum matching that is unstable. Preference lists are shown alongside each individual, with partners shaded. People on the left in a preference list are more preferred A maximum matching that is stable Maximum vs maximal stable matchings Maximum vs maximal stable matchings Augmenting 7-paths in 2 different configurations. Ties are indicated with parentheses Eliminating a cycle by spinning flow. At each node we increment along an incoming edge and decrement along an outgoing edge or vice-versa. Change in allocation at each node is Feasible allocation x for edge (m i, w j ) and 0 < x ij < Path of length 5 in the half-integral auxiliary graph G. Becomes and augmenting path if diagonal edges are chosen in the matching LP and IP solutions to the instance I Instance where allocation to men in ties is equal. The fractional edges form a cycle An instance where continuous rounding fails Augmenting 3-path Two examples of augmenting 5-paths eliminated due to GS-LP algorithm The unique configuration of an augmenting 5-path Augmenting 7-paths due to 2 different configurations Different types of preference lists. G and O indicate the GS-LP and optimal edges. T indicates a tie, and (p) indicates that the particular person has high priority at the end of GS-LP Invalid Augmenting 5-path Edge belonging to M M OP T. indicates a tied region viii

10 Chapter 1 Introduction Matching is a well-studied problem that finds relevance in several real world scenarios. A set of edges, in which no two share a common node represents a matching. Generally we pair nodes based on some criterion. Matchings can be categorized into two subclasses. The first subclass explores the optimization side of the problem, the objective being to obtain matchings of minimum cost or maximum value. For example, given a set of jobs and machines, and a cost function for all the job-machine pairs, our goal might be to obtain a matching of all jobs to machines at the lowest cost possible. The second subclass models problems that exhibit a game-theoretic flavor. Human organ transplant exchange markets, housing allocation, and stable matchings are some of the more popular examples. Problems from this subclass are also referred to as ordinal matching problems, since they involve ranked preference lists rather than explicit costs. Ordinal matching problems are common in situations where we are unable to assess the quality of the solution in terms of numeric cost. In this thesis we study ordinal matching problems on bipartite graphs, the focus being stable matchings. Interest in stable matchings was sparked in the early 50 s by the initial question of how to better match students to colleges. Wide applicability and relevance has contributed to an increase in attention to this area of study from mathematicians, computer scientists, and economists alike in the last few decades. Countries such as the USA, Canada, Scotland, and Japan rely on centralized matching schemes to match graduating medical students to hospitals based on the preferences of students and hospitals for each other [1, 2, 3, 18, 40]. Other relevant applications include matching students to schools [4, 5, 44] and universities [9]. In each of these examples, the participants submit 1

11 w 2 w 1 m 2 m 1 m 2 w 2 m1 m2 w 2 w 1 m 1 w 1 Blocking pair (m 2, w 2 ) Maximum matching M = {(m 2, w 1 ), (m 1, w 2 )} Figure 1.1: A maximum matching that is unstable. Preference lists are shown alongside each individual, with partners shaded. People on the left in a preference list are more preferred. their preferences to a central matching agency. Thus, stable matching is a popular model for ordinal centralized matching in practice. Stable matching belongs to an area of research that has come to be known as algorithmic game theory. The last few years has seen an explosive growth in research being conducted in this area due to its ability to capture various aspects of practical problems. The Nobel prize in economics for the year 2012 was awarded to Alvin E. Roth and Lloyd S. Shapley for their work on stable allocations and practice of market design. Game theoretically speaking, we would like to design a matching mechanism which creates a conducive environment such that given a matching, none of the participants involved have any incentive to break away and trade among themselves in order to obtain a better match. A matching achieving this objective is deemed stable. 1.1 Stable Matchings Stable matchings are represented by a bipartite graph G = (V = L R, E). Conventionally, the sets L and R represent the set of men and women participating in the matching. Each person has a totally ordered (complete) preference list over all the persons from the other set. The set of edges E represents the acceptable pairs, defined by the preference lists of men and women. We define M(p) to be the partner of person p in the matching M. For an edge (m, w) E, where m L and w R, woman w is present in man m s preference list and vice-versa. For some matching M E, edge (m, w) E \ M is a blocking pair if either m is unmatched or prefers w to his partner M(m), and similarly w is unmatched or prefers m to her partner M(w). Simply put, both of them have 2

12 w 2 w 1 m 2 m 1 m 2 w 2 m1 m2 w 2 w 1 m 1 w 1 Stable matching M = {(m 1, w 1 ), (m 2, w 2 )} Figure 1.2: A maximum matching that is stable. worse partners and would be better off matched with each other. Thus, such pairs tend to gain by breaking away from the matching scheme as there exist possibilities to obtain better partners. Any matching M is said to be stable if there exist no blocking pairs. Maximum cardinality matchings need not necessarily be stable. This is illustrated by Figures 1.1 and 1.2. The size of the maximum matching for the instance is two. Figure 1.1 is a maximum matching (indicated by dashed lines) that is unstable due to the presence of blocking pair (m 2, w 2 ), while Figure 1.2 is a maximum stable matching The Gale-Shapley Algorithm Gale and Shapley [14] in a seminal paper, showed that every instance admits a stable matching, has a cardinality of min( L, R ), and proposed an O(n 2 ) algorithm commonly known as the Gale-Shapley (GS) algorithm to obtain such a matching. Each man has a preference list over all the participating women and vice-versa. The preferences are strict and list persons from the most desired to the least desired. The algorithm consists of a series of proposals by single men to acceptable women. In the worst case, the running time is O( E ), as we might have to go through the entire preference list of all the men. The idea behind the algorithm is simple. All men start out single. An arbitrary single man m is chosen to propose. Man m proposes to the most desirable woman w on his list he has not proposed to yet. On receiving a proposal, woman w has two choices: accept or reject. She accepts the proposal if single or m is a strictly better match than her current partner, and rejects otherwise. Once matched, a woman stays matched as any accepted proposal is only from a strictly better suitor. Each rejection causes a man to move down his list and consider lesser desired woman, while each acceptance enables a woman to move up her list and be matched 3

13 with more desired men. A stable matching thus obtained is man-optimal and woman-pessimal. An interesting observation is that running the GS algorithm could give us different stable matchings, but the same set of people always end up matched. Choosing an arbitrary single man for proposing seems to introduce a level of indeterminism. However, from [14] we see that such indeterminism is inconsequential, as termination of the GS algorithm guarantees a stable matching. 1.2 Stable Matching Generalizations Owing to the number of participants in large matching schemes, it is unrealistic to assume that each participant has a complete preference list. For example, approximately 30,000 medical school graduates take part in a centralized matching scheme; the National Residency Matching Program (NRMP), which matches them to hospitals. It would be unrealistic to expect a hospital to rank all of the applicants. Thus, we look at the three possible generalizations based on modifications to the preference lists of the participants. Stable Matching with Incomplete Preference Lists. We allow incomplete lists where each man ranks a subset of the participating women and vice-versa. Here, a blocking pair also includes pairs (m, w) E \ M such that man m is unmatched and woman w prefers man m to her partner M(w) or vice-versa. Every instance to the new problem admits a stable matching, which can be found using the GS algorithm [18]. The matching thus obtained might not match everyone participating, but it can be shown that in every stable matching, the same set of men and women end up matched [13]. Thus, the size of every stable matching is the same. Stable Matching with Ties. A similar generalization to the problem is to allow preference lists to contain ties (indifference to choices). This commonly occurs in practice as people might have equal preferences for certain choices (e.g. one prefers vanilla and chocolate flavors equally). The definition of stability in this case changes slightly. Three classes of stability have been studied in the literature, namely weak, strong and super. In super stability, a pair (m, w) E \ M is blocking if for man m, woman w is equally or more preferred by m than his current match or vice-versa. In strong stability, a pair (m, w) E \ M is blocking if man m strictly prefers woman w to his current match and woman w prefers man m equally or more than her current match. In weak stability, a 4

14 w 2 Tie m 1 m 2 m 2 w 2 w 2 w 1 m1 m 1 w 1 Maximum stable matching Maximal stable matching Figure 1.3: Maximum vs maximal stable matchings. pair (m, w) E \ M is considered blocking if man m strictly prefers woman w to his match and vice-versa. It is useful to note that any super stable matching is strongly stable, and any strongly stable matching is weakly stable. In case of super and strong stability, there exist instances for which no stable matching exists. Nevertheless, algorithms exist which can in polynomial time determine if an instance admits a matching that is super or strongly stable [21, 27]. A weakly stable matching always exists and can be found in polynomial time [21]. Due to this reason it seems to be by far the more popular of the three classes. Thus, we consider instances under weak stability in this thesis. Here, we can use the GS algorithm, breaking ties arbitrarily and still obtain stable matchings with cardinality min( L, R ). Stable Matching with Ties and Incomplete Lists (SMTI). Following the same line of thought, one could ask what happens if both ties and incomplete lists are allowed. The notation indicates a strict preference between two choices under consideration, while indicates strict preference or indifference. To re-state the definition of weak stability formally, a pair (m, w) E \ M is said to be blocking if the following constraints hold: (i) M(m) w, (ii) w is single or m w M(w), and (iii) m is single or w m M(m). Unlike the previous two generalizations, one can obtain stable matchings with varying cardinalities here. Figure 3.1 is one such example. Here, the size of the maximum cardinality stable matching M = {(m 1, w 1 ), (m 2, w 2 )} is two and is indicated by solid edges. Any matching is said to be maximal if we cannot add any further edges to the already existing matching. In Figure 3.1, the maximal stable matching M = (m 1, w 2 ) has cardinality one and is indicated by a dashed edge. Given this possibility, we would like to obtain stable matchings of maximum cardinality 5

15 (MAX SMTI). Unfortunately, the MAX SMTI problem was shown to be NP-hard [24, 34]. In the special case where ties are restricted one side (women), the problem still remains NP-hard. Definition 1. An α-approximation algorithm for an optimization problem is a polynomial-time algorithm that obtains a ratio of at most α between an optimal solution and any solution given by the algorithm over all the instances of the problem. α is greater than 1 for a maximization problem. A stable matching, being maximal, provides a matching that is at least 1 2 the size of an optimal matching. In other words, an optimal matching has at most twice as many edges as in a maximal matching. Running the GS algorithm on an SMTI instance gives a maximal stable matching. Thus we can observe that the GS algorithm gives a 2-approximation to the MAX SMTI problem. 1.3 Main Contributions In this dissertation, we consider the MAX SMTI problem with ties restricted to one side. The current best known approximation factor is [39]. We improve upon this result to obtain an approximation ratio of In order to obtain the improved result, our algorithm uses information from the linear programming (LP) relaxation of the interger programming (IP) formulation of the SMTI problem to break ties. We analyze the algorithm using another LP, commonly known as a factor revealing LP. Here we look at the residual structure of the matching obtained by the algorithm and rewrite the structure in terms of LP constraints. Solving the new LP gives the necessary approximation ratio. In Chapter 2, we survey the literature and present the current known results. Chapter 3 discusses the preliminaries necessary to understand and analyze the problem. In Chapter 4, we look at common approximation techniques and the reasons they might be inapplicable to the MAX SMTI problem. In Chapter 5, we describe the algorithm and lay the foundations necessary to develop a factor-revealing LP and use it to obtain an approximation ratio , thereby improving upon the current best known ratio. We conclude in Chapter 6 with open questions due to this work. 6

16 Chapter 2 Background The MAX SMTI problem has received substantial attention in the literature. It was shown to be NP-hard [24, 34]. In the last few years, there have been a sequence of improvements over the trivial 2-approximation obtained with the GS algorithm. Initial improvements were obtained using local search techniques [25, 27]. The approximation ratio here approaches 2 as V tends to infinity. The factor was further improved via clever enhancements to 15 8 [26]. Kiraly [30] introduced the idea of promotions to break ties and further improved the approximation ratio to 5 3. He used a modification of the GS algorithm such that each man is now permitted to propose to the women on his list more than once, each time with higher priority. McDermid [35] improved this to 3 2 by exploiting a classical graph theoretic result on maximum bipartite matching known as Gallai- ( ) Edmonds decomposition. The runtime complexity of this algorithm is O V 3 2 E. Currently, this is the best known result for the general SMTI problem where ties are allowed in the preference lists of both men and women. Paluch [38] and Király [31] gave linear time algorithms while obtaining the same ratio. The only known approximation ratio better than for the general case is 7. Huang et al. [20] obtained this bound for the special case where the length of each tie is restricted to 2 and ties are allowed on both sides. In the special case where ties are restricted to one side (i.e., ties allowed in the preference lists of either men or women but not both), the problem continues to be NP-hard. For this case, Iwama et al. [28] succeeded in improving the approximation ratio to They formulated the maximum cardinality stable matching problem as an integer program, and relied on the fractional values obtained by solving its linear programming relaxation to break ties along with the idea of 7

17 promotions used earlier by Kiraly [30] and McDermid [35]. The integrality gap (which we describe later) of the SMTI LP relaxation for one-sided ties is lower bounded by e [28]. Thus, we cannot obtain better approximation bounds than this using the SMTI LP relaxation. Recently, Huang et al. [20] improved approximation ratio to They start by computing a half-integral stable matching using a modification of the GS algorithm. Following this, they use a charging argument in order to obtain the result. Unlike the previous approaches, here each man is allowed two proposals and each woman can accept up to two proposals. The analysis of the algorithm was further improved by Radnai [39] and the bound was reduced to Currently, this is the best known approximation bound. Inapproximability results determine the bounds beyond which we cannot improve upon the approximation ratios. They are usually tied to two conjectures, namely (i) P NP, and (ii) unique games conjecture (UGC). Obtaining an approximation ratio better than the bound would imply the resolution of the conjectures. The general MAX SMTI problem cannot be approximated to better than than 4 3 unless P = NP [46]. Under the much stronger UGC, it cannot be approximated to better [46]. For the special case where the length of a man s preference list is bounded by 2, the problem is polynomial-time solvable even when women s preference lists are incomplete and contain ties. On increasing the length of a man s preference list to 3, the problem becomes NP-hard [22]. If we relax our problem and allow ties only on one side, we can approximate better than the general case. The problem remains NP-hard, and we cannot obtain a lower bound better than 21 19, unless P = NP [46]. Under the UGC, we cannot obtain a better lower bound than 5 4 [19]. For the special case where ties are permitted at the end of preference lists in one-sided ties, Iwama et al. [23] were able to match the UGC lower bound. We cannot improve upon the results of McDermid [35] for cases where ties are allowed on both the sides given the UGC. Thus, a natural question to raise would be: whether we can improve upon the current best known result by [39] of for the case where we restrict ties to one side. In this dissertation, we answer this question in the affirmative and show that this can be improved to Stable Allocation. The stable matching problem is a one-to-one matching problem, as each man can be matched to at most one woman. The NRMP program has been in use since the 1950s and is a many-to-one generalization where each medical student is matched to a hospital and the participating hospitals can accept many students. Here, we can think of each individual to be unit- 8

18 sized, while hospitals can be seen as non-unit-sized. Baïou and Balinski [6] went a step further and looked at the case where both sides of the bipartite graph consisted of non-unit-sized entities. Each such entity has a quota q(i), which indicates the number of elements from the other side it should be matched with. Later, they extended this and studied the case where quotas q(i) are arbitrary real numbers [7]. In literature, this has also been known as the ordinal transportation problem, as it is similar to the cost-based transportation problem. Dean et al. [12] gave a clever algorithm to solve the problem in O( E log V ) time improving upon the previously known O( V E ) time. These generalizations can be seen as high-multiplicity variants of the stable matching problem. We show that the new approach due to Huang et al. [20] can be considered more naturally in the context of stable allocations, and that this leads to an alternative mathematical framework that may be useful to consider moving forward for developing improved approximations to the SMTI problem. Factor Revealing LPs. We analyze our algorithm with the help of another LP. For purposes of analysis, we are interested in an instance that produces the worst possible approximation factor. We formulate the new LP such that its corresponding polytope captures all possible instances that could come into existence due to the algorithm. Now, we can have the objective function represent the approximation factor of the algorithm for the problem. The task of computing the approximation factor is reduced to finding an instance with the worst approximation factor in the new LP polytope. Such a technique has come to be known as a factor revealing LP. Similar LPs have been used on several occasions [17, 33, 36], before being formally known as they are today. The best known ratio for the facility location problem is due to a combination of factor revealing LPs and other techniques [29]. Of late, this idea has found relevance in several online bipartite matching problems including [37, 15, 16]. Similarly, this technique was used to obtain tighter bounds for the greedy remote-clique problem [8] and secretary problems [10, 11]. 9

19 Chapter 3 Preliminaries In this chapter, we take a look at the preliminaries helpful in understanding and analyzing the MAX SMTI problem with one-sided ties. In particular, we will be looking at two important relaxations for the SMTI problem based on (i) LP relaxation, and (ii) stable allocation relaxation. 3.1 Symmetric Difference and Augmenting Paths Definition 2. Symmetric Difference. For two graphs G and G with common vertex set V, the symmetric difference G G is the graph containing all the edges that appear in exactly one of G or G. For a stable matching instance, let M OP T be an optimal (i.e., maximum cardinality) stable matching and M be any maximal stable matching. If we take the symmetric difference of M M OP T, we get a graph with paths and even length cycles. The edges on any path or cycle alternate between edges from M OP T and M. Definition 3. Augmenting Path. We call an odd length alternating path with start and end edges belonging to M OP T an augmenting path, since along this path, we could always exchange the edges belonging to matching M with edges belonging to matching M OP T. The size of matching M obtained after this transformation is greater than the M by 1. 10

20 w 2 Tie m 1 m 2 m 2 w 2 w 2 w 1 m1 m 1 w 1 Maximum stable matching Maximal stable matching Figure 3.1: Maximum vs maximal stable matchings. Note that in the context of stable matching, the result M may not be stable. Alternating paths of even length do not affect the size of our matching when toggled this way. As augmenting paths are of odd length, the number of edges in it can be expressed by 2k + 1 for k 1. For an augmenting (2k + 1)-path, we have k + 1 edges from M OP T and k edges from M. Lemma 4. Let graph G = M OP T M. If G does not contain any augmenting (2k 1)-paths or shorter, then M OP T k+1 k M. Proof. The symmetric difference does not contain any augmenting (2k 1)-paths or shorter. As a result, the length of the augmenting paths in G is at least (2k + 1). The augmenting paths are disjoint. For an augmenting path, toggling edges in M to edges in M OP T increases the size of our matching by 1. This implies that the difference between M OP T and M is the number of disjoint augmenting paths p (i.e., M OP T M = p). The number of M OP T edges is at least p(k + 1) (i.e., M OP T p(k+1)) as we know that the length of augmenting paths is at least (2k+1). Substituting for p in the above equation, we get M OP T M = p M OP T k+1. Solving this, we can establish that M OP T k+1 k M Augmenting 3-paths and their Elimination The graph M M OP T does not contain any augmenting 1-paths, as this would block the matching M. The worst approximation ratio we can obtain is when M M OP T consists solely of augmenting 3-paths (paths of length three). Figure 3.1 depicts an augmenting 3-path. The solid edges belong to M OP T while the dashed edge belongs to M. Here, for every edge from M, we have two edges from M OP T. Thus we obtain an approximation ratio of 2. Figure 3.2 is an example where we 11

21 w 4 m 4 w 4 m 3, m 4 w 4 m 4 w 4 m 3, m 4 w 3, w 4 m 3 w 3 m 2, m 3 w 3, w 4 m 3 w 3 (m 2 = m 3 ) w 2, w 3 m 2 w 2 (m 1 = m 2 ) w 3, w 2 m 2 w 2 m 2, m 1 w 2, w 1 m 1 w 1 m 1 w 2, w 1 m 1 w 1 m 1 I II Figure 3.2: Augmenting 7-paths in 2 different configurations. Ties are indicated with parentheses. assume that there exist no augmenting 3 or augmenting 5-paths in the symmetric difference. Hence, we are left with augmenting paths of length 7 and longer. In case of augmenting 7-paths, 3 of the edges come from M, while the remaining 4 edges come from M OP T. Thus, the worst approximation factor we could get here would be 4 3. For the general case, McDermid [35] was able to find a solution M for which M M OP T contained no augmenting 3-paths, giving a 3 2 approximation. The runtime complexity of this algorithm was O( V 3 2 E ). Paluch [38] and Király [31] were able to reduce the runtime complexity to O( E ) while achieving the same approximation ratio. Király [30] introduced the idea of promotions. For the restricted case with one-sided ties, this is powerful enough to prevent any augmenting 3-paths from coming into existence. We start with proposals from single men as in the GS algorithm. Each time a man exhausts his preference list and ends up unmatched, we promote him to the next priority level and let him propose from the start of his list. Initially, each man starts at priority level zero. If a man reaches level i, this implies that he has exhausted his list i times. We run this until all the men are matched or are single and have reached level 2. In case of ties, women accept proposals from men who are at a strictly higher level than others in the tie. Doing so eliminates any augmenting 3-paths in M M OP T. For purposes of contradiction, suppose we end up with an augmenting 3-path as in Figure 3.1. Here m 2 is single, and has reached level 2 implying that he must have proposed to w 2 after being promoted to level 1. On the other hand m 1 never exhausted his list as he is matched to w 2 and w 1 on his list is single. He remains at level 0 till the end. Thus it is contradictory to assume that w 2 accepted a proposal from m 1 at level 0 instead of a proposal from m 2 at level at least 2. 12

22 Though useful, this technique fails to eliminate all augmenting 5-paths as we shall see in the next chapter. Iwama et al. [28], Huang et al. [20] and Radnai [39] removed a constant fraction of augmenting 5-paths for the restricted case with one-sided ties. This resulted in approximation ratios of 25 17, and 28. Further, Iwama et al. [23] were able to eliminate augmenting paths of length 3, 5 and 7 for the one-sided case where ties are allowed to occur at the end of a preference list. Thus, by Lemma 4 their algorithm obtains an approximation ratio of 5 4 for this special case. 3.2 Linear Programming Relaxation The following integer program (IP) is a generalization of [43] and [41] for the MAX SMTI problem with one-sided ties. For each (m i, w j ) man-woman pair, we introduce a variable x ij. The variable takes a value 1 if (m i, w j ) pair is in the matching and 0 otherwise. The set E consists of edges (m i, w j ) based on the preference lists. Sets L and R, consist of participating men and women respectively. Any matching obtained solving the IP will be an optimal stable matching. Objective max (m i,w j) E x ij (3.1) Constraints x ij 1 m i L (3.2) x ij 1 m i w j R (3.3) x ij + x i j 1 w j mi w j m i wj m i (m i, w j ) E (3.4) x ij {0, 1} (m i, w j ) E (3.5) Constraints 3.2 and 3.3 indicate that each person can be matched to at most one other person. We will refer to constraint 3.4 as the stability constraint. This constraint states that for every edge (m i, w j ) E, at least one of m i or w j should be matched to a partner that is equally or more preferred than along this edge, in line with the definition of weak stability. This ensures that both m i and w j are not matched to someone worse than each other and thus helps retain stability. 13

23 If the stability constraint is satisfied with equality, it is said to be tight. We now state the linear programming (LP) relaxation of the aforementioned IP for sake of completeness. In the relaxation, we remove the restriction that variables x ij take integral values. Thus our LP solution might contain x ij values that are fractional. Objective max (m i,w j) E x ij (3.6) Constraints x ij 1 m i L (3.7) x ij 1 m i w j R (3.8) x ij + x i j 1 w j mi w j m i wj m i (m i, w j ) E (3.9) x ij 0 (m i, w j ) E (3.10) Let the optimal solution obtained by solving the LP be OP T LP and the optimal integral solution be OP T IP. The value of OP T LP may or may not be integral. This implies that each person may be matched completely or fractionally. We know that OP T LP OP T IP, since the optimal IP solution is still feasible for the LP relaxation. A general technique applied in obtaining an approximation algorithm is to cleverly round the fractional LP solution to obtain a feasible integral solution. In doing so, if the objective does not change by much, we can achieve good approximation factors. Unfortunately, the integrality gap (see Definition 5) of the LP relaxation has been shown to be lower bounded by e [28]. Thus, this is the best approximation ratio we can hope to achieve by rounding the LP. Definition 5. The integrality gap of an integer program is the worst-case ratio over all instances of the problem of value of an optimal solution to the linear programming relaxation (LP) to value of an optimal solution to the integer programming formulation [45]. 14

24 3.2.1 Integrality Gap Example For sake of completeness, we present here the instance which gives the e integrality gap [28]. Consider the following instance I 1. m 1 : w 1 w k w k+1 w 1 : (m 1 m k ) m k+1 m k : w 1 w k w 2k w k : (m 1 m k ) m 2k m k+1 : w 1 w k+1 : m 1 m 2k : w k w 2k : m k The integral optimal solution OP T IP = {(m i, w i ) i = 1,..., k} has size k. But its corresponding optimal fractional solution OP T LP has the following solution x. x ij = 1 k ( 1 1 k ) j 1 for (i, j) {1, 2,..., k} {1, 2,..., k} x i,k+i = ( 1 1 k ) k for i = {1, 2,..., k} x k+i,i = 1 ( 1 1 k ) i 1 Substituting these, we obtain the optimal LP value to be k + k ( 1 1 k ) k. The integrality gap is given by OP T LP /OP T IP which is equal to 1 + ( 1 1 k ) k. This expression tends to e to infinity. as k tends Algorithms using LP Relaxation Solving the SMTI LP (see Section 3.2) gives us a solution that is optimal in the fractional sense. The solution does not necessarily translate to a stable matching that is integral. Iwama et al. [28] were the first to use the x ij values obtained by solving the SMTI LP relaxation to break ties while running the GS algorithm to produce an integer stable matching. All the approaches prior to this, promoted men only after they exhausted their entire preference list and remained single. 15

25 The promotion scheme here is more sensitive. On being rejected, a man is promoted immediately, though fractionally. As before, in case of a tie, a woman accepts a proposal from a man at a strictly higher priority level. Based on LP properties, they were able to show that a constant fraction of augmenting 5-paths are removed due to GS-LP. This enabled them to break the 3 2 barrier and reduce the ratio to Stable Allocation Relaxation Recent techniques used to improve the approximation ratio for the SMTI problem have similar characteristics to that of the stable allocation problem. Here, we show how to interpret stable matchings in the framework of stable allocations as this may be of use in developing new techniques to improve the approximation factor for the SMTI problem. The stable allocation problem is generally studied in the context of jobs and machines. The jobs belong to the set L and the machines belong to the set R. Each job i has a processing time p i, each machine has a capacity c j and at most u ij = min(p i, c j ) units of job i can be assigned to machine j along edge (i, j). The jobs have a strict, transitive and complete preference list over the machines and vice-versa. The goal is to seek an allocation that is stable. We denote by x ij the real-valued allocation along the edge (i, j). An edge is considered blocking if the following condidtions hold: (i) x ij < u ij, (ii) x ij > 0 and j i j, and (iii) x i j > 0 and i j i. The conditions indicate that it is possible to increase allocation along the more preferred edge (i, j), and thus the given allocation cannot be stable. An allocation is a feasible stable allocation if the following conditions are satisfied: x ij = p i, i L (3.11) j R x ij = c j, j R (3.12) i L p i x ij c j j ij i ji x i j = 0, (i, j) E (3.13) x ij 0 (i, j) E (3.14) We can redefine the constraints in the context of stable matching by changing p i s and c j s to 16

26 1 giving an alternative relaxation of the stable matching problem, which we call the SA relaxation. We will be switching back to the m i, w j terminology used in the context of stable matchings. We now state the stable matching constraints as the stable allocation constraints. A matching is stable and feasible if the following conditions hold x ij 1 m i L (3.15) w j R m i L 1 x ij 1 w j R (3.16) w j mi w j x ij 1 m i wj m i x i j = 0 (m i, w j ) E (3.17) x ij 0 (m i, w j ) E (3.18) Constraints 3.15 and 3.16 are exactly the matching constraints. Constraint 3.17 indicates that for an edge (m i, w j ), either m i or w j are matched to someone equally preferred or better. Thus, constraint 3.17 ensures stability in the matching as at least one of m i or w j is matched to someone at least as good as the other. The SA relaxation is indeed a relaxation to the SMTI problem: restricting x variables to integers models exactly the SMTI problem, just like the LP. Hence its optimum is an upper bound on OP T IP. The SA allocation exhibits the following interesting property. If the allocation up to w j on m i s list is 1, then the allocation up to m i on w j s list may or may not add up to 1. Given an SMTI instance I, we can saturate the nodes by adding dummy nodes to both sides of the bipartite graph. Let the sets be L D and R D. The dummy nodes appear last in the preference list. Any allocation to a dummy node from a regular node implies that the regular node is unsaturated. Let the function f(x) = x ij. i / L D,j / R D Lemma 6. Let x be a fractional solution to the SA relaxation. We can round it to a feasible integral solution x for SMTI such that f(x ) f(x ). Proof. Consider an instance I of the SMTI problem. We modify the instance and add dummy nodes on both sides of the bipartite graph such that all the men and women can be saturated. Each dummy node appears at the end of preference lists of members from the opposite set. Now, any unsaturated 17

27 +ɛ m 2 w 2 ɛ ɛ m 1 w 1 +ɛ Edge-weight 1 2 Figure 3.3: Eliminating a cycle by spinning flow. At each node we increment along an incoming edge and decrement along an outgoing edge or vice-versa. Change in allocation at each node is 0. man/woman can be saturated as they are matched to the dummy partner. Thus, constraints 3.11 and 3.12 are satisfied with equality (due to the dummy nodes catering to the partially matched nodes). We start with a feasible solution x to the SA relaxation. Now, consider the graph G = {(m i, w j ) 0 < x ij < 1}. If G contains no edges, we have an integral solution. Otherwise, due to our construction G contains a cycle as all the nodes are fully allocated. We can eliminate a cycle by spinning flow in clockwise or anti-clockwise direction. After choosing a direction to spin flow, we do the following at each node: increase allocation along the incoming edge and decrease allocation along the outgoing edge. Note that the edges chosen for this are fractional due to the definition of graph G. Figure 3.3 is an example of spinning flow across one such cycle. As we increment and decrement by the same amount, the corresponding allocation at the node remains unchanged. The chosen direction should not decrease f(x), as this would imply an increase in allocation to dummy nodes. We can eliminate cycles by shifting allocation in this manner. We continue this till we have eliminated all cycles. What we are left with at the end is an integral solution. Let f(x ) be the new allocation after the elimination of a cycle. We have f(x ) f(x ) as we never choose a direction to spin flow that increases allocation to the dummy nodes. At the same time, the increment and decrement at a node only occur along fractional edges. This conserves the allocation at each node. Consider the initial stable allocation x and an edge (m i, w j ) G. Assume that x i j = m i wj m i 1 (shown in Figure 3.4). Thus we have x i j = 0 due to Constraint For purposes of m i wj m i contradiction, suppose that after eliminating a cycle the allocation we have < 1. We m i wj m i x i j know that the increment-decrement does not result in loss of allocation at a node. This implies that the loss in region r 1 was compensated by the gain in region r 2 on w j s list (see Figure 3.4). This 18

28 w j = 1 m i w j m i r 1 r 2 Figure 3.4: Feasible allocation x for edge (m i, w j ) and 0 < x ij < 1. is not possible as there are no edges in G from region r 2. Thus, increment-decrement happens in region r 1 and the new allocation x is still feasible. As we do not lose any increase allocation to dummy nodes in the process of spinning allocation, we have f(x ) f(x ). If x is an optimal solution for the SA relaxation, then due to the fact that it is a relaxation, we have f(x ) OP T IP. Since Lemma 6 allows us to round x to a feasible integer solution with no decrease in f(x ), this implies the reverse inequality as well, so f(x ) = OP T IP ; otherwise stated: Corollary 7. The SA relaxation has no integrality gap, so approximating the SMTI is equivalent to approximating its SA equivalent New Approach by Huang et al. Huang et al. [20] recently gave an algorithm that has a stable allocation interpretation. Unlike the GS algorithm, here each person is allowed to be matched with up to two persons from the other set. As a result, each man has 2 proposals to make and each woman can accept up to 2 proposals. Based on the algorithm, each edge (m i, w j ) E is given a value as follows: x ij = 1 if w j accepts both of m i s proposals, x ij = 1 2 if w j accepts exactly one proposal from m i and x ij = 0 if w j received or accepts no proposals from m i. Using the x ij values thus obtained, they create an auxiliary graph G such that (m i, w j ) G if x ij > 0. This is equivalent to computing a half-integral matching. The construction restricts the maximum degree of G to 2. They compute a maximum matching in G such that all nodes with degree two are matched. This can be done quickly as G consists of disjoint paths and cycles. They show that augmenting 3-paths never come into existense. Due to the construction, no augmenting 5-paths can exist independently, are dependent on other augmenting structures of length 5 and beyond for its existence. Figure 3.5 is an example of a path of length 5 present independently in the auxiliary graph G. If rounded incorrectly (diagonal edges 19

29 w 3 m 3 w 3 m 2 m 3 w 2 w 3 m 2 w 2 (m 1 m 2 ) w 2 w 1 m 1 w 1 m 1 Edge-weight 1 2 Figure 3.5: Path of length 5 in the half-integral auxiliary graph G. Becomes and augmenting path if diagonal edges are chosen in the matching. chosen over horizontal edges), this results in an augmenting 5-path. Given this fractional allocation, we can see that it satisfies constraints 3.15 to Finding a maximum matching in which all degree 2 nodes are matched translates to rounding a fractional allocation to an integral one. By Lemma 6, we know that we can round a fractional stable allocation to an integral one without any loss. The maximum matching on G gives edges (m 1, w 1 ), (m 2, w 2 ) and (m 3, w 3 ). We started out with a fractional matching of weight 5 2 and rounded it up to 3. Thus, independent paths of length 5 are eliminated. Let M be the matching obtained by running the Huang et al. [20] algorithm and M OP T an optimal matching for the instance under consideration. Let R be the set of all augmenting 5-paths in M M OP T and Q = M M OP T \ R. The set Q now consists of augmenting paths of length 2l + 3 7, cycles with at least 2l edges or alternating paths with 2l 1 edges with l edges from M. As paths in R are not independent, they show a mapping scheme in which each path p R is mapped to some augmenting path in Q. In turn, using a novel charging scheme, they show that each matched node (from matching M) in Q can be charged at most 1.5 units and that in an augmenting path q Q, at most 2l q nodes can be charged. Thus, each such path can be charged at most c q 3l q units. Thus, we have M OP T = ( M OP T q + 3c q ), where we are considering 3 q Q edges from each of the mapped c q augmenting 5-paths. Similarly, M = ( M q + 2c q ). The approximation ratio is maximized by charging augmenting 7-paths for augmenting 5-paths. Thus we have l q = 2. Maximizing over M OP T M q Q using the expressions from before, we get the necessary approximation ratio of Recently, Radnai [39] improved their analysis by showing that at most 5.5 units can be charged to an augmenting 7-path and reduced the bound further to Currently, this is the best known approximation bound. 20

30 Chapter 4 Counterexamples and Lessons Learned This chapter presents negative results and counterexamples to help streamline future research efforts by avoiding investigation of avenues of attack that may not seem promising. As a reminder, we are trying to design algorithms for the MAX SMTI problem with one-sided ties. 4.1 Iterative and Randomized Rounding The first approach considered is commonly known as iterative rounding. The procedure utilizes properties of extreme point solutions obtained from solving LPs. Any LP can be visualized as optimization over a convex polytope. Polyhedral theory states that optimal solutions to the LP are obtained at extreme points (corners) of the polytope. In its most common form, we start by solving the LP under consideration. Based on some inherent property of the extreme points in the polytope (e.g. we might be able to prove that some variable in an extreme point solution must be equal or close to 1), the value of a variable x is set to 1. Next, we solve the LP with the additional constraint x = 1. We do this iteratively, setting one variable at a time to 1 till we end up with an integral solution. Singh [42] utilized properties of extreme point solutions in order to obtain iterative algorithms with good approximation factors for problems such as the bounded degree spanning tree problem [32]. For example, suppose one could show that for our problem, there exists an edge 21