In the Multi-Spreader Crane Scheduling Problem (MSCSP), containers with identical dimensions but ... more In the Multi-Spreader Crane Scheduling Problem (MSCSP), containers with identical dimensions but variable weights are arranged along a grid. A multispreader crane is used to lift all the containers. The crane has m > 1 modes. When it is in the pth mode, the crane can remove p adjacent containers along the same row at the same time as long as the total weight of the containers does not exceed the loading capacity κ p. Such a lift takes h p minutes. It also takes c p,q minutes for the crane to switch from mode p to q when p = q. The goal is to find a crane lift sequence so that the total time it takes to lift all the containers is minimized. This paper investigates the computational complexity of MSCSP. First, we establish a connection between greedy crane lift sequences and supersequences. We then prove that MSCSP is NP-hard when the crane has three or more modes by a reduction from a version of the Shortest Common Supersequence problem. Lastly, we investigate two problems that arise naturally when heuristics are used to solve MSCSP. We show that one can be solved using dynamic programming while the other remains computationally hard. We also provide an approximation algorithm that behaves nicely when the changeover times are not much larger than the lifting times of the crane.
Third International Conference on Quality Software, 2003. Proceedings., 2003
In this paper, we consider a problem that arises in black box testing: generating small test suit... more In this paper, we consider a problem that arises in black box testing: generating small test suites (i.e., sets of test cases) where the combinations that have to be covered are specified by input-output parameter relationships of a software system. That is, we only consider combinations of input parameters that affect an output parameter. We also do not assume that the input parameters have the same number of values. To solve this problem, we revisit the greedy algorithm for test generation and show that the size of the test suite it generates is within a logarithmic factor of the optimal. Unfortunately, the algorithm's main weaknesses are its time and space requirements for construction. To address this issue, we present a problem reduction technique that makes the greedy algorithm and possibly any other test suite generation method more efficient if the reduction in size is significant.
Let I be a stable matching instance with N stable matchings. For each man m, order his N stable p... more Let I be a stable matching instance with N stable matchings. For each man m, order his N stable partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N , not only is αi a matching, it is also stable. The αi's are called the generalized median stable matchings of I. In this paper, we present a new characterization of these stable matchings that is solely based on I's rotation poset. We then prove the following: when i = O(log n), where n is the number of men, αi can be found efficiently; but when i is a constant fraction of N , finding αi is NPhard. We also consider what it means to approximate the median stable matching of I, and present results for this problem.
In this paper, we describe the operation of barter trade exchanges by identifying key techniques ... more In this paper, we describe the operation of barter trade exchanges by identifying key techniques used by trade brokers to stimulate trade and satisfy member needs, and present algorithms to automate some of these techniques. In particular, we develop algorithms that emulate the practice of trade brokers by matching buyers and sellers in such a way that trade volume is maximized while the balance of trade is maintained as much as possible. We show that the buyer/seller matching and trade balance problems can be decoupled, permitting efficient solution as well as numerous options for matching strategies. exchange rule of thumb that maximizing single-period trade volume while maintaining balance of trade helps to maximize trade volume over the long run.
The hospitals/residents (HR) problem is a many-to-one generalization of the stable marriage (SM) ... more The hospitals/residents (HR) problem is a many-to-one generalization of the stable marriage (SM) problem. Researchers have been interested in variants of stable matchings that either satisfy a set of additional contraints or are optimal with respect to some cost function. In this paper, we show that broad classes of feasibility and optimization stable matching problems in the HR setting can be solved efficiently provided certain tasks (such as checking the feasibility of a stable matching or computing the cost of a stable matching) can also be done efficiently. To prove our results, we make use of an HR instance's meta-rotation poset to explore its stable matchings. An algorithm that can discover all the meta-rotations of the instance serves as a starting point for all our algorithms.
Electronic Commerce Research and Applications, 2005
In this paper, we describe the operation of barter trade exchanges by identifying key techniques ... more In this paper, we describe the operation of barter trade exchanges by identifying key techniques used by trade brokers to stimulate trade and satisfy member needs, and present algorithms to automate some of these techniques. In particular, we develop algorithms that emulate the practice of trade brokers by matching buyers and sellers in such a way that trade volume is maximized while the balance of trade is maintained as much as possible. We model the trade balance problem as a minimum cost circulation problem (MCC) on a network. When the products have uniform cost or when the products can be traded in fractional units, we solve the problem exactly. Otherwise, we present a novel stochastic rounding algorithm that takes the fractional optimal solution to the trade balance problem and produces a valid integer solution. We then make use of a greedy heuristic that attempts to match buyers and sellers so that the average number of suppliers that a buyer must use to satisfy a given product need is minimized. We present results of empirical evaluation of our algorithms on test problems and on simulations built using data from an operating trade exchange.
A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the... more A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the colors is the identity map. It is proper-distinguishing if the coloring is both proper and distinguishing. The distinguishing number of G, D(G), is the smallest integer k so that G has a distinguishing k-coloring; the distinguishing chromatic number of G, χ D (G), is defined similarly. It has been shown recently that the distinguishing number of a planar graph can be determined efficiently by counting a related parameter-the number of inequivalent distinguishing colorings of the graph. In this paper, we demonstrate that the same technique can be used to compute the distinguishing number and the distinguishing chromatic number of an interval graph. We make use of PQ-trees, a classic data structure that has been used to recognize and test the isomorphism of interval graphs; our algorithms run in O(n 3 log 3 n) time for graphs with n vertices. We also prove a number of results regarding the computational complexity of determining a graph's distinguishing chromatic number.
An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a funct... more An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences α and α generated by non-trivial walks on G starting at u, f (α) = f (α ) if and only if the two walks end at the same node. The function f is referred to as a coding function of λ. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G) ≥ ∆ + (G), where ∆ + (G) is the maximum outdegree of G. Let us say that a function τ : We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem "Given G, does G have a WSD-labeling that uses a particular coding function f ?" is NP-complete even when G and f are fairly simple. Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D)) = ∆ + (H(D)) = d * , where d * is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D)) is ∆ + (H(D)). Our WSD-labelings are poset-based, making use of Birkhoff's characterization of distributive lattices and Dilworth's theorem for posets.
Let I be a stable matching instance with N stable matchings. For each man m, order his (not neces... more Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as p i (m). Let α i consist of the man-woman pairs where each man m is matched to p i (m). Teo and Sethuraman proved this surprising result: for i = 1 to N , not only is α i a matching, it is also stable. The α i 's are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem. In this paper, we present a new characterization of the generalized median stable matchings that provides interesting insights. It implies that the generalized median stable matchings in the middle-α (N +1)/2 when N is odd, α N/2 and α N/2+1 when N is even-are fair not only in a local sense but also in a global sense because they are also medians of the lattice of stable matchings. We then show that there are some families of SM instances for which computing an α i is easy but that the task is NP-hard in general. Finally, we also consider what it means to approximate a median stable matching and present results for this problem.
Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Give... more Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Given a declustering scheme D , its response time with respect to a query Q , rt ( Q ), is defined to be the maximum number of data blocks of the query stored by the scheme in any one of the disks. If | Q | is the number of data blocks in Q and M is the number of disks, then rt ( Q ) is at least ⌈| Q |/ M ⌉. One way to evaluate the performance of D with respect to a set of range queries Q is to measure its additive error ---the maximum difference of rt ( Q ) from ⌈| Q |/ M ⌉ over all range queries Q ∈ Q.In this article, we consider the problem of designing declustering schemes for uniform multidimensional data arranged in a d -dimensional grid so that their additive errors with respect to range queries are as small as possible. It has been shown that for a fixed dimension d ≥ 2, any declustering scheme on an M d grid, a grid with length M on each dimension, will always incur an additive erro...
We present two hardness results on the man-exchange stable marriage problem, one of which settles... more We present two hardness results on the man-exchange stable marriage problem, one of which settles a recent conjecture of Irving on the complexity of determining whether a given instance of the stable marriage problem with short preference lists has a man-exchange stable matching.
For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are... more For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are stable matchings that assign each participant to his or her (lower/upper) median stable partner. Moreover, for SM instances, a stable matching has this property if and only if it is a median of the distributive lattice formed by the instance's stable matchings. In this paper, we show that the above local/global median phenomenon first observed in SM stable matchings also extends to SR stable matchings because SR stable matchings form a median graph. In the course of our investigations, we also prove that three seemingly different structures are pairwise duals of each other -median graphs give rise to mirror posets and vice versa, and mirror posets give rise to SR stable matchings and vice versa. Together, they imply that for every median graph G, there is an SR instance I(G) whose graph of stable matchings is isomorphic to G. Our results are analogous to the pairwise duality results known for distributive lattices, posets, and SM stable matchings. Interestingly, they can also be inferred from the work of Feder in the early 1990's. Our constructions and proofs, however, are smoother generalizations of those used for SM instances.
A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labe... more A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph. 1
In this paper, we consider the problem of counting and sampling structures in graphs. We define a... more In this paper, we consider the problem of counting and sampling structures in graphs. We define a class of "edge universal labeling problems"---which include proper $k$-colorings, independent sets, and downsets---and describe simple algorithms for counting and uniformly sampling valid labelings of graphs, assuming a path decomposition is given. Thus, we show that several well-studied counting and sampling problems are fixed parameter tractable (FPT) when parameterized by the pathwidth of the input graph. We discuss connections to counting and sampling problems for distributive lattices and, in particular, we give a new FPT algorithm for exactly counting and uniformly sampling stable matchings.
Proceedings of the 22nd ACM Conference on Economics and Computation, 2021
It is well known that every stable matching instance I has a rotation poset R(I) that can be comp... more It is well known that every stable matching instance I has a rotation poset R(I) that can be computed efficiently and the downsets of R(I) are in one-to-one correspondence with the stable matchings of I. Furthermore, for every poset P, an instance I(P) can be constructed efficiently so that the rotation poset of I(P) is isomorphic to P. In this case, we say that I(P) realizes P. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al.[1] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions. Our goal is to characterize the rotation posets that arise from these models: k-bounded, where each agent has at most k acceptable partners; k-attribute, where e...
Birkhoff's fundamental theorem on distributive lattices states that for every distributive lattic... more Birkhoff's fundamental theorem on distributive lattices states that for every distributive lattice L there is a poset P L whose lattice of down-sets is order-isomorphic to L. Let G(L) denote the cover graph of L. In this paper, we consider the following problems: Suppose we are simply given P L. How do we compute the eccentricity of an element of L in G(L)? How about a center and the radius of G(L)? While eccentricity, center and radius computations have long been studied for various classes of graphs, our problems are different in that we are not given the graph explicitly; instead, we only have a structure that implicitly describes the graph. By making use of the comparability graph of P L , we show that all the said problems can be solved efficiently. One of the important implications of these results is that a center stable matching, a kind of fair stable matching, can be computed in polynomial time.
A graph G is an NG-graph if χ(G) + χ(G) = |V (G)| + 1. We characterize NG-graphs solely from degr... more A graph G is an NG-graph if χ(G) + χ(G) = |V (G)| + 1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on n vertices, allows us to compute the sizes of each of these classes. Finally, we provide a bijection between unbalanced split graphs on n vertices and split graphs on n − 1 or fewer vertices providing evidence for our conjecture that the rapid growth in the number of split graphs comes from the balanced split graphs.
In the Multi-Spreader Crane Scheduling Problem (MSCSP), containers with identical dimensions but ... more In the Multi-Spreader Crane Scheduling Problem (MSCSP), containers with identical dimensions but variable weights are arranged along a grid. A multispreader crane is used to lift all the containers. The crane has m > 1 modes. When it is in the pth mode, the crane can remove p adjacent containers along the same row at the same time as long as the total weight of the containers does not exceed the loading capacity κ p. Such a lift takes h p minutes. It also takes c p,q minutes for the crane to switch from mode p to q when p = q. The goal is to find a crane lift sequence so that the total time it takes to lift all the containers is minimized. This paper investigates the computational complexity of MSCSP. First, we establish a connection between greedy crane lift sequences and supersequences. We then prove that MSCSP is NP-hard when the crane has three or more modes by a reduction from a version of the Shortest Common Supersequence problem. Lastly, we investigate two problems that arise naturally when heuristics are used to solve MSCSP. We show that one can be solved using dynamic programming while the other remains computationally hard. We also provide an approximation algorithm that behaves nicely when the changeover times are not much larger than the lifting times of the crane.
Third International Conference on Quality Software, 2003. Proceedings., 2003
In this paper, we consider a problem that arises in black box testing: generating small test suit... more In this paper, we consider a problem that arises in black box testing: generating small test suites (i.e., sets of test cases) where the combinations that have to be covered are specified by input-output parameter relationships of a software system. That is, we only consider combinations of input parameters that affect an output parameter. We also do not assume that the input parameters have the same number of values. To solve this problem, we revisit the greedy algorithm for test generation and show that the size of the test suite it generates is within a logarithmic factor of the optimal. Unfortunately, the algorithm's main weaknesses are its time and space requirements for construction. To address this issue, we present a problem reduction technique that makes the greedy algorithm and possibly any other test suite generation method more efficient if the reduction in size is significant.
Let I be a stable matching instance with N stable matchings. For each man m, order his N stable p... more Let I be a stable matching instance with N stable matchings. For each man m, order his N stable partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N , not only is αi a matching, it is also stable. The αi's are called the generalized median stable matchings of I. In this paper, we present a new characterization of these stable matchings that is solely based on I's rotation poset. We then prove the following: when i = O(log n), where n is the number of men, αi can be found efficiently; but when i is a constant fraction of N , finding αi is NPhard. We also consider what it means to approximate the median stable matching of I, and present results for this problem.
In this paper, we describe the operation of barter trade exchanges by identifying key techniques ... more In this paper, we describe the operation of barter trade exchanges by identifying key techniques used by trade brokers to stimulate trade and satisfy member needs, and present algorithms to automate some of these techniques. In particular, we develop algorithms that emulate the practice of trade brokers by matching buyers and sellers in such a way that trade volume is maximized while the balance of trade is maintained as much as possible. We show that the buyer/seller matching and trade balance problems can be decoupled, permitting efficient solution as well as numerous options for matching strategies. exchange rule of thumb that maximizing single-period trade volume while maintaining balance of trade helps to maximize trade volume over the long run.
The hospitals/residents (HR) problem is a many-to-one generalization of the stable marriage (SM) ... more The hospitals/residents (HR) problem is a many-to-one generalization of the stable marriage (SM) problem. Researchers have been interested in variants of stable matchings that either satisfy a set of additional contraints or are optimal with respect to some cost function. In this paper, we show that broad classes of feasibility and optimization stable matching problems in the HR setting can be solved efficiently provided certain tasks (such as checking the feasibility of a stable matching or computing the cost of a stable matching) can also be done efficiently. To prove our results, we make use of an HR instance's meta-rotation poset to explore its stable matchings. An algorithm that can discover all the meta-rotations of the instance serves as a starting point for all our algorithms.
Electronic Commerce Research and Applications, 2005
In this paper, we describe the operation of barter trade exchanges by identifying key techniques ... more In this paper, we describe the operation of barter trade exchanges by identifying key techniques used by trade brokers to stimulate trade and satisfy member needs, and present algorithms to automate some of these techniques. In particular, we develop algorithms that emulate the practice of trade brokers by matching buyers and sellers in such a way that trade volume is maximized while the balance of trade is maintained as much as possible. We model the trade balance problem as a minimum cost circulation problem (MCC) on a network. When the products have uniform cost or when the products can be traded in fractional units, we solve the problem exactly. Otherwise, we present a novel stochastic rounding algorithm that takes the fractional optimal solution to the trade balance problem and produces a valid integer solution. We then make use of a greedy heuristic that attempts to match buyers and sellers so that the average number of suppliers that a buyer must use to satisfy a given product need is minimized. We present results of empirical evaluation of our algorithms on test problems and on simulations built using data from an operating trade exchange.
A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the... more A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the colors is the identity map. It is proper-distinguishing if the coloring is both proper and distinguishing. The distinguishing number of G, D(G), is the smallest integer k so that G has a distinguishing k-coloring; the distinguishing chromatic number of G, χ D (G), is defined similarly. It has been shown recently that the distinguishing number of a planar graph can be determined efficiently by counting a related parameter-the number of inequivalent distinguishing colorings of the graph. In this paper, we demonstrate that the same technique can be used to compute the distinguishing number and the distinguishing chromatic number of an interval graph. We make use of PQ-trees, a classic data structure that has been used to recognize and test the isomorphism of interval graphs; our algorithms run in O(n 3 log 3 n) time for graphs with n vertices. We also prove a number of results regarding the computational complexity of determining a graph's distinguishing chromatic number.
An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a funct... more An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences α and α generated by non-trivial walks on G starting at u, f (α) = f (α ) if and only if the two walks end at the same node. The function f is referred to as a coding function of λ. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G) ≥ ∆ + (G), where ∆ + (G) is the maximum outdegree of G. Let us say that a function τ : We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem "Given G, does G have a WSD-labeling that uses a particular coding function f ?" is NP-complete even when G and f are fairly simple. Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D)) = ∆ + (H(D)) = d * , where d * is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D)) is ∆ + (H(D)). Our WSD-labelings are poset-based, making use of Birkhoff's characterization of distributive lattices and Dilworth's theorem for posets.
Let I be a stable matching instance with N stable matchings. For each man m, order his (not neces... more Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as p i (m). Let α i consist of the man-woman pairs where each man m is matched to p i (m). Teo and Sethuraman proved this surprising result: for i = 1 to N , not only is α i a matching, it is also stable. The α i 's are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem. In this paper, we present a new characterization of the generalized median stable matchings that provides interesting insights. It implies that the generalized median stable matchings in the middle-α (N +1)/2 when N is odd, α N/2 and α N/2+1 when N is even-are fair not only in a local sense but also in a global sense because they are also medians of the lattice of stable matchings. We then show that there are some families of SM instances for which computing an α i is easy but that the task is NP-hard in general. Finally, we also consider what it means to approximate a median stable matching and present results for this problem.
Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Give... more Declustering schemes allocate data blocks among multiple disks to enable parallel retrieval. Given a declustering scheme D , its response time with respect to a query Q , rt ( Q ), is defined to be the maximum number of data blocks of the query stored by the scheme in any one of the disks. If | Q | is the number of data blocks in Q and M is the number of disks, then rt ( Q ) is at least ⌈| Q |/ M ⌉. One way to evaluate the performance of D with respect to a set of range queries Q is to measure its additive error ---the maximum difference of rt ( Q ) from ⌈| Q |/ M ⌉ over all range queries Q ∈ Q.In this article, we consider the problem of designing declustering schemes for uniform multidimensional data arranged in a d -dimensional grid so that their additive errors with respect to range queries are as small as possible. It has been shown that for a fixed dimension d ≥ 2, any declustering scheme on an M d grid, a grid with length M on each dimension, will always incur an additive erro...
We present two hardness results on the man-exchange stable marriage problem, one of which settles... more We present two hardness results on the man-exchange stable marriage problem, one of which settles a recent conjecture of Irving on the complexity of determining whether a given instance of the stable marriage problem with short preference lists has a man-exchange stable matching.
For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are... more For stable marriage (SM) and solvable stable roommates (SR) instances, it is known that there are stable matchings that assign each participant to his or her (lower/upper) median stable partner. Moreover, for SM instances, a stable matching has this property if and only if it is a median of the distributive lattice formed by the instance's stable matchings. In this paper, we show that the above local/global median phenomenon first observed in SM stable matchings also extends to SR stable matchings because SR stable matchings form a median graph. In the course of our investigations, we also prove that three seemingly different structures are pairwise duals of each other -median graphs give rise to mirror posets and vice versa, and mirror posets give rise to SR stable matchings and vice versa. Together, they imply that for every median graph G, there is an SR instance I(G) whose graph of stable matchings is isomorphic to G. Our results are analogous to the pairwise duality results known for distributive lattices, posets, and SM stable matchings. Interestingly, they can also be inferred from the work of Feder in the early 1990's. Our constructions and proofs, however, are smoother generalizations of those used for SM instances.
A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labe... more A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph. 1
In this paper, we consider the problem of counting and sampling structures in graphs. We define a... more In this paper, we consider the problem of counting and sampling structures in graphs. We define a class of "edge universal labeling problems"---which include proper $k$-colorings, independent sets, and downsets---and describe simple algorithms for counting and uniformly sampling valid labelings of graphs, assuming a path decomposition is given. Thus, we show that several well-studied counting and sampling problems are fixed parameter tractable (FPT) when parameterized by the pathwidth of the input graph. We discuss connections to counting and sampling problems for distributive lattices and, in particular, we give a new FPT algorithm for exactly counting and uniformly sampling stable matchings.
Proceedings of the 22nd ACM Conference on Economics and Computation, 2021
It is well known that every stable matching instance I has a rotation poset R(I) that can be comp... more It is well known that every stable matching instance I has a rotation poset R(I) that can be computed efficiently and the downsets of R(I) are in one-to-one correspondence with the stable matchings of I. Furthermore, for every poset P, an instance I(P) can be constructed efficiently so that the rotation poset of I(P) is isomorphic to P. In this case, we say that I(P) realizes P. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al.[1] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions. Our goal is to characterize the rotation posets that arise from these models: k-bounded, where each agent has at most k acceptable partners; k-attribute, where e...
Birkhoff's fundamental theorem on distributive lattices states that for every distributive lattic... more Birkhoff's fundamental theorem on distributive lattices states that for every distributive lattice L there is a poset P L whose lattice of down-sets is order-isomorphic to L. Let G(L) denote the cover graph of L. In this paper, we consider the following problems: Suppose we are simply given P L. How do we compute the eccentricity of an element of L in G(L)? How about a center and the radius of G(L)? While eccentricity, center and radius computations have long been studied for various classes of graphs, our problems are different in that we are not given the graph explicitly; instead, we only have a structure that implicitly describes the graph. By making use of the comparability graph of P L , we show that all the said problems can be solved efficiently. One of the important implications of these results is that a center stable matching, a kind of fair stable matching, can be computed in polynomial time.
A graph G is an NG-graph if χ(G) + χ(G) = |V (G)| + 1. We characterize NG-graphs solely from degr... more A graph G is an NG-graph if χ(G) + χ(G) = |V (G)| + 1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on n vertices, allows us to compute the sizes of each of these classes. Finally, we provide a bijection between unbalanced split graphs on n vertices and split graphs on n − 1 or fewer vertices providing evidence for our conjecture that the rapid growth in the number of split graphs comes from the balanced split graphs.
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Papers by Christine Cheng