ports fast insertions and accesses, with a fairly high space- utilization. Typically, dictionaries are used in combination with a hashing scheme, which is a mapping from the items to their locations in the table. A common assumption in theoretical work is that such mappings are random. Under this assumption, the connection between hashing and our setting is as follows: if an item has d randomly chosen locations where it can be inserted into the table, then the problem of inserting a series of items into the table reduces to the problem of finding a **matching** in a random **bipartite** **graph** in an online manner.

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According to the experimental results, our method per- formance (90% recall with 93% precision) is similar to (and even better than) the approach proposed in [26] but it is done without preprocessing of the target video. However, subse- quence identification results may be highly dependent on the **testing** material, which is usually scarce and not especially representative. Moreover, choosing an appropriate feature that enhances performance of a **matching** algorithm is not a trivial task. Therefore, as future work, we will consider more relevant/robust descriptors and study their impact on the precision and recall rates.

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The **Matching** Theorem. The fact we’ve just discovered is the crucial step in proving the **Matching** Theorem; from here it’s easy, as follows.
Consider a **bipartite** **graph** with an equal number of nodes on the left and right, and suppose it has no **perfect** **matching**. Let’s take a maximum **matching** in it — one that includes as many edges as possible. Since this **matching** is not **perfect**, and since there are an equal number of nodes on the two sides of the **bipartite** **graph**, there must be a node W on the right-hand side that is unmatched. We know there cannot be an augmenting path containing W , since then we’d be able to enlarge the **matching** — and that isn’t possible since we chose a **matching** of maximum size. Now, by our previous claim, since there is no augmenting path beginning at W , there must be a constricted set containing W . Since we’ve deduced the existence of a constricted set from the fact that the **graph** has no **perfect** **matching**, this completes the proof of the **Matching** Theorem.

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Can we bypass the Isolation Lemma altogether and deterministically isolate minimum weight solutions in speciﬁc sit- uations? Recent results illustrate that one may be able to use the structure of speciﬁc computational problems under consideration to achieve non-trivial deterministic isolation. In [6], the authors used the structure of directed paths in planar graphs to prescribe a simple weight function that is computable deterministically in logarithmic space with respect to which the minimum weight directed path between any two vertices is unique. In [7], the authors isolated a **perfect** **matching** in planar **bipartite** graphs. In [8], the authors give a generalized weight function that isolates directed paths in planar graphs and **perfect** matchings in undirected **bipartite** planar graphs by bypassing the use of grid graphs altogether. In this paper we extend the deterministic isolation technique of [7,8] to isolate a minimum weight **perfect** **matching** in **bipartite** graphs embedded on constant genus surfaces. This is more interesting in light of the fact that for constant genus graphs even the existence of a polynomially bounded weight function, that isolates a minimum weight **perfect** **matching**, was not known earlier. As a future direction it would be interesting to consider the general **bipartite** **graph** K n , n , and prove the existence of

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1.4 Reduction to Maximum Flow
Given a **bipartite** **graph** partitioned into vertex sets, create a new **graph** by adding vertex s with edges from s to each vertex in one partition, and adding vertex t with edges from t to each vertex in the other partiiton. If we give each edge unit weight, then solving the maximum flow problem on this **graph** also solves the maximal **matching** problem on the original **graph**, if we ignore the new edges.

A **perfect** **matching** in a **graph** is a set of edges such that every vertex is incident with exactly one edge in this set. So if a **graph** has a **perfect** **matching**, then it has an even number of vertices. In this paper we only consider graphs with an even number of vertices. The **matching** preclusion number of a **graph** G, denoted by mp(G), is the minimum number of edges whose deletion leaves the resulting **graph** without **perfect** matchings. Any such optimal set is called an optimal **matching** preclusion set. We note that mp(G) = 0 if G has no **perfect** matchings. This concept of **matching** preclusion was introduced by Brigham et al. [3] and further studied in [9, 6] as a measure of robustness in the event of edge failures in interconnection networks, as well as a theoretical connection to conditional connectivity, “changing and unchanging of invariants” and extremal **graph** theory. We refer the readers to [3] for details and additional references. The idea of studying the effect of deleting edges on maximum matchings has been considered prior to [3]. For example, Hung et al. [13] studied the most “vital” edges of a **matching** in a **bipartite** **graph**, that is, those edges whose individual removal results in the largest decrease of the objective function value in the corresponding weighted **matching** problem. It turns out that this problem can be obtained from the dual solution (by linear programming) as observed by Volgenant [23] in reference of [15]. More recently, Zenklusen [25] studied **matching** interdiction and [21, 26] studied d-blockers of a **graph**. In particular, a d-blocker is a set of edges whose deletion decreases the cardinality of a maximum **matching** by at least d. So a 1-blocker corresponds to a **matching** preclusion set if the underlying **graph** has a **perfect** **matching**. Algorithmic aspects of finding optimal 1-blockers have been considered by Boros et al. [2].

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Lemma 3.2 Let the adversary present a k-regular **bipartite** **graph**. If the advice is such that the edges in the **matching** are not determined univocally in k different time steps, then the algorithm could end up with a non-**perfect** **matching**. Proof. Let us assume that there are k different choices for which the advice is not enough, i.e., the algorithm may choose a wrong edge/vertex (since there will be at least two edges with the same advice). If there are k wrong choices, the adversary may present a vertex incident to exactly those vertices that were

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provide a 3 2 -approximation algorithm for this problem.
Stable matchings
Suppose we are given a **bipartite** **graph** and for each vertex a linear preference order of its neighbors. A **matching** M is not stable if there is an edge vw not in M such that v prefers w and w prefers v to their respective partners in M . The well-known algorithm by Gale and Shapley yields a stable **matching** in polynomial time [14]. It is known that any two stable matchings cover the same vertices, so the stable matchings are **perfect** matchings of some subgraph. Furthermore, they form a distributive lattice under rotations on preference- oriented cycles, see for example [14]. Essentially, the symmetric difference of two stable matchings consists of disjoint cycles (of several lengths) and we may exchange edges on these cycles to obtain another stable **matching**. If we drop the preferences, then the question is simply if we can find a transformation between two **perfect** matchings by exchanging edges on cycles in the symmetric difference. Clearly the answer is always yes, for example by processing the cycles in the symmetric difference one by one. We consider a similar setting, but restrict the length of the cycles.

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In this paper discussed about topologized graphs referred to Antoine Vella [1]. In 2005 Antoine Vella [1] tried to express combinatorial concepts in topological language. As a part the investigation classical topology, pre path, path [2], pre cycle, compact space, cycle space and bond space [3], locally connectedness and ferns were defined. Given a hypergraph H [4], the classical topology on ∪ is the collection of all sets U such that, if U contains a vertex v, then it also contains all hyperedges incident with v. It is interesting to note that all these topologies are either defined on the vertex set, or on the union of vertex set and edge set. For Topological **graph** theory, more important application can be found in printing electronic circuits where the aim is to print (embed) a circuit on a circuit board without two connections crossing each other and resulting in a short circuit. And the **Bipartite** graphs are extensively used in modern coding theory, especially to decode code words received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner **graph** is a **bipartite** **graph** in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors. In this paper discussed about some new methods of topology for **bipartite** graphs.

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an O(h)-competitive deterministic algorithm for MPMD on tree metrics, and used it to show an O(log n)-competitive algorithm for general metrics. Additionally, they provided a lower bound of Ω √ log n on the competitive ratio of randomized algorithms for MPMD.
Another strand of research in the economics and operations literatures studied **matching** with delays in stochastic and more structural environments. Anderson et al. [3] and Ashlagi et al. [4] study a model with an underlying stochastic **graph** and assume agents arrive according to some process. They seek to minimize agents’ average waiting time and find that greedy **matching** is asymptotically optimal. Akbarpour et al. [2] allow for agents departures and find that when departure times are known, greedy **matching** leads to a suboptimal match rate. These papers do not have the notion of distance; agents only care about when they match and not whom they match to, which is key to the fact that greedy **matching** performs well. Baccara et al. [6] look at a two-sided market where on each side agents can be of one of two types and one type is of higher “quality” than the other. They assume a single agent on each side arrives every time period and find that the optimal **matching** policy accumulates agents up to a certain threshold.

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on 714,396,886 processors. Obviously, these numbers are based on the bigO complexity functions and thus do not provide us with exact values. However, they are presented to show the practical absurdity of a perfectly reasonable theoretical result. Not only the most powerful existing computer has fewer than 10000 processors and the largest number of processors existing ever in a single machine was about 65000, but also one should ask how reasonable are thecomplexity functions involving 714 million of processors as far as, for instance, their connectivity and communication are concerned. Finally, observe how small a **graph** how large a computer are required and try to extrapolate the required computational power for realistic sizes of the networks for which flow problems are considered in practice.

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1. There is no node in X which has more than one outgoing edge where there is a flow.
2. There is no node in Y which has more than one incoming edge where there is a flow.
3. The number of edges between X and Y which carry flow is k.
By these observations, it is straightforward to conclude that the set of edges carrying flow in f forms a **matching** of size k for the **graph** G. Likewise, given a **matching** of size k in G, we can construct a flow of size k in G 0 . Therefore, solving for maximum flow in G 0 gives us a maximum **matching** in G. Note that we used the fact that when edge capacities are integral, Ford-Fulkerson produces an integral flow.

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We propose a lexical similarity approach to grade the similarity of two sentences, where a maximal weighted **bipartite** match is found be- tween the tokens of the two sentences. The ap- proach is robust enough to apply across different datasets. The results on the STS test datasets are encouraging to say the least. The tokens are sin- gle word tokens in case of the first system, while in the second system, named and monetary en- tities, percentages, dates and times are handled too. A token-token similarity measure is integral to the approach and we use both a statistical sim- ilarity measure and a WordNet based word sim- ilarity measure for the same. In the final run of the task, apart from capturing the aforemen- tioned entities, we heuristically extract adjecti- vally and numerically modified words. Also, the last run naively attempts to capture the context around the tokens using grammatical dependen- cies, which in turn is used to measure context similarity.

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j x ij ≤ 1) in which case we merely increase x ij as much as possible given the degree constraint.
However, this “stateless balancing” algorithm fails to be better than 2-competitive. To construct a counterexample, we take the example form Section 4.1 and blow up each vertex into n vertices, carefully modifying the edge set to cause the algorithm to make catastrophic decisions. The set L now has 2n vertices, which we will label as a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n , and the set R has 2n vertices labeled c 1 , c 2 , . . . , c n , d 1 , d 2 , . . . , d n . Each vertex c j has n + 1 neighbors: it is connected to a j and also to b 1 , b 2 , . . . , b n . Each vertex d j has only one neighbor, namely b j . The maximum **matching** in this **graph** has size 2n: it matches (a i , c i ) and (b i , d i ) for i = 1, . . . , n. If the vertices c 1 , . . . , c n , d 1 , . . . , d n arrive in that order, the stateless balancing algorithm will first assign a value of n+1 1 to each edge incident to c 1 , . . . , c n . Thus, when d 1 , . . . , d n start arriving, each of them has a unique neighbor and the fractional degree of that neighbor is already n+1 n , so d j can contribute only n+1 1 additional units to the size of the fractional **matching**. Thus, when the algorithm is finished processing the entire **graph**, the total size of its fractional **matching** is n + n+1 n , only slightly more than half of the optimum.

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j x ij ≤ 1) in which case we merely increase x ij as much as possible given the degree constraint.
However, this “stateless balancing” algorithm fails to be better than 2-competitive. To construct a counterexample, we take the example form Section 4.1 and blow up each vertex into n vertices, carefully modifying the edge set to cause the algorithm to make catastrophic decisions. The set L now has 2n vertices, which we will label as a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n , and the set R has 2n vertices labeled c 1 , c 2 , . . . , c n , d 1 , d 2 , . . . , d n . Each vertex c j has n + 1 neighbors: it is connected to a j and also to b 1 , b 2 , . . . , b n . Each vertex d j has only one neighbor, namely b j . The maximum **matching** in this **graph** has size 2n: it matches (a i , c i ) and (b i , d i ) for i = 1, . . . , n. If the vertices c 1 , . . . , c n , d 1 , . . . , d n arrive in that order, the stateless balancing algorithm will first assign a value of n+1 1 to each edge incident to c 1 , . . . , c n . Thus, when d 1 , . . . , d n start arriving, each of them has a unique neighbor and the fractional degree of that neighbor is already n+1 n , so d j can contribute only n+1 1 additional units to the size of the fractional **matching**. Thus, when the algorithm is finished processing the entire **graph**, the total size of its fractional **matching** is n + n+1 n , only slightly more than half of the optimum.

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Results. Weighted **bipartite** **graph** G = ( V , E ) with a cost function ce : E ® { 0 , 1 } , that associates each edge with one of two possible values (e.g. 0 or 1) is considered. Maximum **matching** in the **graph** in new setting consists in finding among all matchings containing maximum number of edges with weight 1, one having maximal cardinality. Two algorithms with complexity O ( m n ) being modified versions of the Hopcroft-Karp algo- rithm are proposed. Examples of using these algorithms for removing gaps of lines and finding true correspondence of minutiae in fingerprint recognition are considered.

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The main motivation for our interest in the problem we consider in this paper, given the aforementioned desirable properties of the VCG mechanism, is to find frameworks to encode unit-demand auctions that are expressive enough to have suitable applications while being restrictive enough to yield efficient algorithms for finding VCG outcomes. For instance, consider a unit-demand auction for last-minute vacation packages in which some trusted third party (e.g., TripAdvisor) assigns a “quality” rating for each package and each bidder formulates a unit-demand bid for every package by simply declaring a linear function of the qualities of packages, i.e., determining the intercept and slope of this linear function. Within this context, we can formulate an auction as a complete weighted **bipartite** **graph** in the family that we consider in this paper. In some of the popular auction sites, e.g., eBay, bidding takes place in multiple rounds. eBay implements a variant of an English auction to sell a single item; the bids are sealed, but the second highest bid (plus one small bid increment), which is the amount that the winner pays, is displayed throughout the auction. We employ a similar approach by accepting the bids one-by-one and by maintaining an efficient representation of the tentative outcome for the enlarged set of bids. We show that we can process each bid in ˜ O( √ n) time where n denotes the number of items in the auction.

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Abstract. Necessary and Sufficient conditions are given for a fuzzy **graph** on a cycle or a complete **graph** to be a square **perfect** fuzzy **matching**. As a consequence, it is shown that at a particular condition a square **perfect** fuzzy **matching** is not a (2, ) regular fuzzy **graph**.

tative **matching** on gender-role attitudes. In their absence, **perfect** assortative **matching** would unfold, and the model would then collapse to the standard unitary model.
The setup of the model is organized as follows. First, a simple model of intrahousehold bargaining over time allocations is introduced, where the balance of power between the spouses is itself a function of their time allocation decisions. After deriving the intra- household equilibrium as function of partners’ preferences, I then move to the marriage market, which is populated by a continuum of men and women di↵ering on their pref- erences. Anticipating the future intrahousehold outcome as a function of their own and their potential partners’ preferences, each individual chooses a partner to maximize their expected marital payo↵. In order to be able to empirically allow for independent variation in partners’ preferences, I then explicitly allow for mismatch to occur in equilibrium due to search frictions. Finally, I derive implications of the model that will guide the empirical analysis.

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Popular matchings in WCHA
5.1 Introduction
In this chapter, we extend our study of popular matchings from Chapter 4 to the Weighted Capacitated House Allocation problem with no ties (WCHA), a generalisation of CHA in which each agent has an associated weight that indicates his priority. The assignment of weights to agents allow us to build up a spectrum of priority levels for agents in the competition for houses in situations where the total capacity of the houses is less than the number of agents. In turn, this gives some agents a better chance of “doing well”. For instance, the assignment of weights can enable DVD rental companies like Amazon to give priority to those members who have paid more for privileged status whenever a certain title is limited in stock. Alternatively, weights may be assigned to candidates in job markets based on objective criteria such as academic results or relevant work experience. The main results of this chapter, and their organisation are as follows. In Section 5.2, we first develop necessary conditions for a **matching** to be popular in a WCHA instance I. In Section 5.3, we identify a structure in the underlying **graph** of I that singles out those edges that cannot belong to a popular **matching**. We then use these two results in conjunction to construct a O( √ Cn 1 + m) time algorithm for finding a maximum popular

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