Boolean formulae, hypergraphs and combinatorial topology

With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex. In particular, the set of satisfiable formulae in k-conjunctive normal form with ?n variables has the homotopy type of Θ(Cube(n,n−k)), where Cube(n,n−k) is a hypergraph associated to the (n−k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give calculations and examples for other hypergraphs as well. Indeed studying the theta complex of hypergraphs is an interesting problem in its own right.     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

An interpretation of the Dittert conjecture in terms of semi-matchings

Let G be K_n,n with non-negative edge weights and let U and V be the two colour classes of vertices in G. We define a k-semimatching in G to be a set of k edges such that the edges either have distinct ends in U or distinct ends in V. Semimatchings are to be counted according to the product of the weights on the edges in the semimatching. The Dittert conjecture is a longstanding open problem involving matrix permanents. Here we show that it is equivalent to the following assertion: For a fixed total weight, the number of n-semimatchings in G is maximised by weighting all edges of G equally. We also introduce sub-Dittert functions which count k-semimatchings and are analogous to the subpermanent functions which count k-matchings. We prove some results about the extremal values of our sub-Dittert functions, and also that the Dittert conjecture cannot be disproved by means of unweighted graphs.     

Rectilinear line segment intersection,layered segment trees,and dynamization

The aim of the present paper is to provide an efficient solution to the following problem: “Given a family of n rectilinear line segments in two-space report all intersections in the family with a query consisting of an arbitrary rectilinear line segment.” We provide an algorithm which takes O(nlog₂n) preprocessing time, o(nlog₂n) space and O(log₂n + k) query time, where k is the number of reported intersections. This solution serves to introduce a powerful new data structure, the layered segment tree, which is of independent interest. Second it yields, by way of recent dynamization techniques, a solution to the on-line version of the above problem, that is the operations INSERT and DELETE and QUERY with a line segment are allowed. Third it also yields a new nonscanning solution to the batched version of the above problem. Finally we apply these techniques to the problem obtained by replacing “line segment” by “rectangle” in the above problem, giving an efficient solution in this case also.     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

The Variational Capacity with Respect to Nonopen Sets in Metric Spaces

We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R ⁿ it is shown to coincide with the usual variational capacity considered in the literature. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.     

Boolean functions with MacWilliams duality

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree n?k for any nonnegative integer k and n ≥ N with some positive integer N under a certain condition.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

On the uniform edge-partition of a tree

We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k?n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k?4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k-split with ratio at most three in O(nlogk) time. Experimental results on random trees are also shown.     

The minimum spanning strong subdigraph problem is fixed parameter tractable

A digraph D is strong if it contains a directed path from x to y for every choice of vertices x,y in D. We consider the problem (MSSS) of finding the minimum number of arcs in a spanning strong subdigraph of a strong digraph. It is easy to see that every strong digraph D on n vertices contains a spanning strong subdigraph on at most 2n−2 arcs. By reformulating the MSSS problem into the equivalent problem of finding the largest positive integer k≤n−2 so that D contains a spanning strong subdigraph with at most 2n−2−k arcs, we obtain a problem which we prove is fixed parameter tractable. Namely, we prove that there exists an O(f(k)n^c) algorithm for deciding whether a given strong digraph D on n vertices contains a spanning strong subdigraph with at most 2n−2−k arcs.We furthermore prove that if k≥1 and D has no cut vertex then it has a kernel of order at most (2k−1)². We finally discuss related problems and conjectures.     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

Reconstructing compositions

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n?3k+1 can be reconstructed from its set of k-deletions, i.e., the set of all compositions of n-k contained in w. As there are compositions of 3k with the same set of k-deletions, this result is best possible.     

The solution to the partition reconstruction problem

Given a partition λ of n, a k-minor of λ is a partition of n−k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k?g(n). In particular, partitions of n?k²+2k are uniquely determined by their sets of k-minors. This result completely solves the partition reconstruction problem and also a special case of the character reconstruction problem for finite groups.     

Worst-case analysis of demand point aggregation for the Euclidean p-median problem

Solving large-scale p-median problems is usually time consuming. People often aggregate the demand points in a large-scale p-median problem to reduce its problem size and make it easier to solve. Most traditional research on demand point aggregation is either experimental or assuming uniformly distributed demand points in analytical studies. In this paper, we study demand point aggregation for planar p-median problem when demand points are arbitrarily distributed. Efficient demand aggregation approaches are proposed with the corresponding attainable worst-case aggregation error bounds measured. We demonstrate that these demand aggregation approaches introduce smaller worst-case aggregation error bounds than that of the honeycomb heuristic [Papadimitriou, C.H., 1981. Worst-case and probabilistic analysis of a geometric location problem. SIAM Journal on Computing 10, 542–557] when demand points are arbitrarily distributed. We also conduct numerical experiments to show their effectiveness.     

Computing a maxian point of a simple rectilinear polygon

Let P be a simple rectilinear polygon with n vertices. There are k points in P. The maxian problem is to locate a single facility in P so as to maximize the sum of its distance from it to the k points. We present an O((n×k)logn) time algorithm for this problem.     

Covering moving points with anchored disks

Consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines. We study the problem of covering all mobile clients using a set of k disks centered at k fixed centers. Each disk exists only at one instant and while it does, covers any client within its coverage radius. The task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk. In particular, we study the optimization problem of minimizing the maximum coverage radius. First we prove that, although the static version of the problem is polynomial, the kinetic version is NP-hard. Moreover, we show that the problem is not approximable by a constant factor (unless P = NP). We then present a generic framework to solve it for fixed values of k, which in turn allows us to solve more general optimization problems. Our algorithms are efficient for constant values of k.     

A 5.875-approximation for the Traveling Tournament Problem

In this paper we propose an approximation for the Traveling Tournament Problem which is the problem of designing a schedule for a sports league consisting of a set of teams T such that the total traveling costs of the teams are minimized. It is not allowed for any team to have more than k home-games or k away-games in a row. We propose an algorithm which approximates the optimal solution by a factor of 2+2k/n+k/(n?1)+3/n+3/(2?k) which is not more than 5.875 for any choice of k≥4 and n≥6. This is the first constant factor approximation for k>3. We furthermore show that this algorithm is also applicable to real-world problems as it produces solutions of high quality in a very short amount of time. It was able to find solutions for a number of well known benchmark instances which are even better than the previously known ones.     

On multiple caps in finite projective spaces

In this paper, we consider new results on (k, n)-caps with n > 2. We provide a lower bound on the size of such caps. Furthermore, we generalize two product constructions for (k, 2)-caps to caps with larger n. We give explicit constructions for good caps with small n. In particular, we determine the largest size of a (k, 3)-cap in PG(3, 5), which turns out to be 44. The results on caps in PG(3, 5) provide a solution to four of the eight open instances of the main coding theory problem for q = 5 and k = 4.     

Reoptimizing the 0-1 knapsack problem

In this paper we study the problem where an optimal solution of a knapsack problem on n items is known and a very small number k of new items arrive. The objective is to find an optimal solution of the knapsack problem with n+k items, given an optimal solution on the n items (reoptimization of the knapsack problem). We show that this problem, even in the case k=1, is NP-hard and that, in order to have effective heuristics, it is necessary to consider not only the items included in the previously optimal solution and the new items, but also the discarded items. Then, we design a general algorithm that makes use, for the solution of a subproblem, of an α-approximation algorithm known for the knapsack problem. We prove that this algorithm has a worst-case performance bound of , which is always greater than α, and therefore that this algorithm always outperforms the corresponding α-approximation algorithm applied from scratch on the n+k items. We show that this bound is tight when the classical Ext-Greedy algorithm and the algorithm are used to solve the subproblem. We also show that there exist classes of instances on which the running time of the reoptimization algorithm is smaller than the running time of an equivalent PTAS and FPTAS.