Constructions of given-depth and optimal multirate rearrangeably nonblocking distributors

Rearrangeable multirate multicast switching networks are customarily called rearrangeable multirate distributors. It has been known for a long time that rearrangeable multirate distributors with cross-point complexity O(nlog?² n) can be constructed, where n is the number of inputs (and outputs) of the switching network. The problem of constructing optimal distributors remains open thus far. This paper gives a general construction of rearrangeable multirate distributors with given depths. One byproduct is a rearrangeable multirate distributor with crosspoint complexity O(nlog?n). We also show that this cross-point complexity is optimal, settling the aforementioned open problem. One of the key ingredients of our new construction is the notion of multirate concentrators. The second ingredient is a multirate version of the Pippenger network. We show how to construct given-depth multirate concentrators and given-depth multirate Pippenger networks with small sizes. When the depth is chosen to optimize the size, we obtain the optimal O(nlog?n) cross-point complexity.     

Largest area convex hull of imprecise data based on axis-aligned squares

In recent years, more and more algorithms related to imprecise data have been proposed. Specifically, some algorithms on computing the maximum area convex hull are designed recently when the imprecise data are modeled as non-overlapping axis-aligned squares or as equal size squares. The time complexity of the best known algorithm based on non-overlapping axis-aligned squares is O(n ⁷). If the squares have equal size and can overlap, the time complexity of the best known algorithm is O(n ⁵). In this paper, we improve the former from O(n ⁷) to O(n ⁵) and improve the latter from O(n ⁵) to O(n ²). These results are obtained by exploiting the non-trivial geometric properties of the problems.     

Faster algorithm to find anti-risk path between two nodes of an undirected graph

For a weighted 2-edge connected graph G=(V,E), we are to find a “minimum risk path” from source s to destination t. This is a shortest s?t path under the assumption that at most one edge on the path may be blocked. The fact that the edge is blocked is known only when we reach a site adjacent to the blocked edge. If n and m are the number of nodes and edges of G, then we show that this problem can be solved in O(n ²) time using only simple data structures. This is an improvement over the previous O(mn+n ²logn) time algorithm. Moreover, with use of more complicated data structures like Fibonacci Heaps and transmuters the time can be further reduced to O(m+nlogn).     

A simplified algorithm for the all pairs shortest path problem with O(n 2logn) expected time

The best known expected time for the all pairs shortest path problem on a directed graph with non-negative edge costs is O(n ²logn) by Moffat and Takaoka. Let the solution set be the set of vertices to which the given algorithm has so far established shortest paths. The Moffat-Takaoka algorithm maintains complexities before and after the critical point in balance, which is the moment when the size of the solution set is n?n/logn. In this paper, we remove the concept of critical point, whereby we make the algorithm simpler and seamless, resulting in a simpler analysis.     

Small grid drawings of planar graphs with balanced partition

In a grid drawing of a planar graph, every vertex is located at a grid point, and every edge is drawn as a straight-line segment without any edge-intersection. It is known that every planar graph G of n vertices has a grid drawing on an (n?2)×(n?2) or (4n/3)×(2n/3) integer grid. In this paper we show that if a planar graph G has a balanced partition then G has a grid drawing with small grid area. More precisely, if a separation pair bipartitions G into two edge-disjoint subgraphs G ₁ and?G ₂, then G has a max?{n ₁,n ₂}×max?{n ₁,n ₂} grid drawing, where n ₁ and n ₂ are the numbers of vertices in G ₁ and?G ₂, respectively. In particular, we show that every series-parallel graph G has a (2n/3)×(2n/3) grid drawing and a grid drawing with area smaller than 0.3941n ² (<(2/3)² n ²).     

A new approximation algorithm for the Selective Single-Sink Buy-at-Bulk problem in network design

The Selective Single-Sink Buy-at-Bulk problem was proposed by Awerbuch and Azar (FOCS 1997). For a long time, the only known non-trivial approach to approximate this problem is the tree-embedding method initiated by Bartal (FOCS 1996). In this paper, we give a thoroughly different approximation approach for the problem with approximation ratio $O(\sqrt{q})$ , where q is the number of source terminals in the problem instance. Our approach is based on a mixed strategy of LP-rounding and the greedy method. When the number q (which is always at most n) is relatively small (say, q=o(log² n)), our approximation ratio $O(\sqrt{q})$ is better than the currently known best ratio O(logn), where n is the number of vertices in the input graph.     

Long cycles in hypercubes with optimal number of?faulty vertices

Let f(n) be the maximum integer such that for every set F of at most f(n) vertices of the hypercube Q _n , there exists a cycle of length at least 2 ⁿ ?2|F| in Q _n ?F. Casta?eda and Gotchev conjectured that $f(n)=\binom{n}{2}-2$ . We prove this conjecture. We also prove that for every set F of at most (n ²+n?4)/4 vertices of Q _n , there exists a path of length at least 2 ⁿ ?2|F|?2 in Q _n ?F between any two vertices such that each of them has at most 3 neighbors in F. We introduce a new technique of potentials which could be of independent interest.     

An almost four-approximation algorithm for maximum weight triangulation

We consider the following planar maximum weight triangulation (MAT) problem: given a set of n points in the plane, find a triangulation such that the total length of edges in triangulation is maximized. We prove an $\Omega(\sqrt{n})$ lower bound on the approximation factor for several heuristics: maximum greedy triangulation, maximum greedy spanning tree triangulation and maximum spanning tree triangulation. We then propose the Spoke Triangulation algorithm, which approximates the maximum weight triangulation for points in general position within a factor of almost four in O(nlog?n) time. The proof is simpler than the previous work. We also prove that Spoke Triangulation approximates the maximum weight triangulation of a convex polygon within a factor of two.     

Assortment optimisation problem: A distribution-free approach

Assortment optimisation is a critical decision that is regularly made by retailers. The decision involves a trade-off between offering a larger assortment of products but smaller inventories of each product and offering a smaller number of varieties with more inventory of each product. We propose a robust, distribution-free formulation of the assortment optimisation problem such that the assortment and inventory levels can be jointly optimised without making specific assumptions on the demand distributions of each product. We take a max-min approach to the problem that provides a guaranteed lower bound to the expected profit when only the mean and variance of the demand distribution are known. We propose and test three heuristic algorithms that provide solutions in O(nlog (n)) time and identify two cases where one of the heuristics is guaranteed to return optimal policies. Through numerical studies, we demonstrate that one of the heuristics performs extremely well, with an average optimality gap of 0.07% when simulated under varying conditions. We perform a sensitivity analysis of product and store demand attributes on the performance of the heuristic. Finally, we extend the problem by including maximum cardinality constraints on the assortment size and perform numerical studies to test the performance of the heuristics.     

Enumerating the edge-colourings and total colourings of a regular graph

In this paper, we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of k-edge-colourings of a connected k-regular graph on n vertices is k?((k?1)!) ^n/2. Our proof is constructive and leads to a branching algorithm enumerating all the k-edge-colourings of a connected k-regular graph in time O ^?(((k?1)!) ^n/2) and polynomial space. In particular, we obtain a algorithm to enumerate all the 3-edge-colourings of a connected cubic graph in time O ^?(2 ^n/2)=O ^?(1.4143 ⁿ ) and polynomial space. This improves the running time of O ^?(1.5423 ⁿ ) of the algorithm due to Golovach et al. (Proceedings of WG 2010, pp. 39–50, 2010). We also show that the number of 4-total-colourings of a connected cubic graph is at most 3?2^3n/2. Again, our proof yields a branching algorithm to enumerate all the 4-total-colourings of a connected cubic graph.     

Computing maximum upward planar subgraphs of single-source embedded digraphs

We show how to compute a maximum upward planar single-source subgraph of a single-source embedded DAG G _φ . We first show that finding a maximum upward planar subgraph of a single-source embedded digraph is NP-complete. We then give a new characterization of upward planar single-source digraphs. We use this characterization to present an algorithm that computes a maximum upward planar single-source subgraph of a single-source embedded DAG. This algorithm takes O(n ⁴) time in the worst case and O(n ³) time on average.     

Some variations on constrained minimum enclosing circle problem

Given a set P of n points and a straight line L, we study three important variations of minimum enclosing circle problem as follows:

1. Computing k identical circles of minimum radius with centers on L, whose union covers all the points in P.
2. Computing the minimum radius circle centered on L that can enclose at least k points of P.
3. If each point in P is associated with one of the k given colors, then compute a minimum radius circle with center on L such that at least one point of each color lies inside it.

We propose three algorithms for Problem (i). The first one runs in O(nklogn) time and O(n) space. The second one is efficient where k?n; it runs in O(nlogn+nk+k ²log³ n) time and O(nlogn) space. The third one is based on parametric search and it runs in O(nlogn+klog⁴ n) time. For Problem (ii), the time and space complexities of the proposed algorithm are O(nk) and O(n) respectively. For Problem (iii), our proposed algorithm runs in O(nlogn) time and O(n) space.     

Min-energy broadcast in mobile ad hoc networks with restricted motion

This paper concerns about energy-efficient broadcasts in mobile ad hoc networks, yet in a model where each station moves on the plane with uniform rectilinear motion. Such restriction is imposed to discern which issues arise from the introduction of movement in the wireless ad hoc networks. Given a transmission range assignment for a set of n stations S, we provide an polynomial O(n ²)-time algorithm to decide whether a broadcast operation from a source station could be performed or not. Additionally, we study the problem of computing a transmission range assignment for S that minimizes the energy required in a broadcast operation. An O(n ³log?n)-time algorithm for this problem is presented, under the assumption that all stations have equally sized transmission ranges. However, we prove that the general version of such problem is NP-hard and not approximable within a (1?o(1))ln?n factor (unless NP?DTIME(n ^O(log?log?n))). We then propose a polynomial time approximation algorithm for a restricted version of that problem.     

Cost sharing and strategyproof mechanisms for set cover games

We develop for set cover games several general cost-sharing methods that are approximately budget-balanced, in the core, and/or group-strategyproof. We first study the cost sharing for a single set cover game, which does not have a budget-balanced mechanism in the core. We show that there is no cost allocation method that can always recover more than $\frac{1}{\ln n}$ of the total cost and in the core. Here n is the number of all players to be served. We give a cost allocation method that always recovers $\frac{1}{\ln d_{\mathit{max}}}$ of the total cost, where d _max is the maximum size of all sets. We then study the cost allocation scheme for all induced subgames. It is known that no cost sharing scheme can always recover more than $\frac{1}{n}$ of the total cost for every subset of players. We give an efficient cost sharing scheme that always recovers at least $\frac{1}{2n}$ of the total cost for every subset of players and furthermore, our scheme is cross-monotone. When the elements to be covered are selfish agents with privately known valuations, we present a strategyproof charging mechanism, under the assumption that all sets are simple sets; further, the total cost of the set cover is no more than ln?d _max times that of an optimal solution. When the sets are selfish agents with privately known costs, we present a strategyproof payment mechanism to them. We also show how to fairly share the payments to all sets among the elements.     

The partition method for poset-free families

Given a finite poset P, let ${\rm La}(n,P)$ denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the n-set, to give simpler new proofs of the known asymptotic behavior of ${\rm La}(n,P)$ , as n→∞, when P is the r-fork $\mathcal {V}_{r}$ , the four-element N poset $\mathcal {N}$ , and the four-element butterfly-poset $\mathcal {B}$ .     

A branch-and-bound algorithm for the minimum cut linear arrangement problem

Given an edge-weighted graph G of order n, the minimum cut linear arrangement problem (MCLAP) asks to find a one-to-one map from the vertices of G to integers from 1 to n such that the largest of the cut values c ₁,…,c _n?1 is minimized, where c _i , i∈{1,…,n?1}, is the total weight of the edges connecting vertices mapped to integers 1 through i with vertices mapped to integers i+1 through n. In this paper, we present a branch-and-bound algorithm for solving this problem. A salient feature of the algorithm is that it employs a dominance test which allows reducing the redundancy in the enumeration process drastically. The test is based on the use of a tabu search procedure developed to solve the MCLAP. We report computational results for both the unweighted and weighted graphs. In particular, we focus on calculating the cutwidth of some well-known graphs from the literature.     

On the mod sum number of H m,n

Let N denote the set of all positive integers. The sum graph G ⁺(S) of a finite subset S?N is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an mod sum graph if it is isomorphic to the sum graph of some S?Z _M \{0} and all arithmetic performed modulo M where M≥|S|+1. The mod sum number ρ(G) of G is the smallest number of isolated vertices which when added to G result in a mod sum graph. It is known that the graphs H _m,n (n>m≥3) are not mod sum graphs. In this paper we show that H _m,n are not mod sum graphs for m≥3 and n≥3. Additionally, we prove that ρ(H _m,3)=m for m≥3, H _m,n ∪ρK ₁ is exclusive for m≥3 and n≥4 and $m(n-1) \leq \rho(H_{m,n})\leq \frac{1}{2} mn(n-1)$ for m≥3 and n≥4.     

Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O ^?(3 ⁿ s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O ^?(2 ⁿ ) for ΠΣΠ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ polynomials with terms of degree ≤2, the first problem cannot be approximated at all for any approximation factor ≥1, nor “weakly approximated” in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-approximation algorithm for ΠΣΠ polynomials with terms of degrees no more a constant λ≥2. On the inapproximability side, we give a n ^(1??)/2 lower bound, for any ?>0, on the approximation factor for ΠΣΠ polynomials. When terms in these polynomials are constrained to degrees ≤2, we prove a 1.0476 lower bound, assuming P≠NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.     

Improved approximation for spanning star forest in dense graphs

A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c∈(0,1), a graph of n vertices is called c-dense if it contains at least cn ²/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ?>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c∈(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.     

An improved time-space lower bound for tautologies

We show that for all reals c and d such that c ² d<4 there exists a positive real e such that tautologies of length n cannot be decided by both a nondeterministic algorithm that runs in time n ^c , and a nondeterministic algorithm that runs in time n ^d and space n ^e . In particular, for every \(d<\sqrt[3]{4}\) there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n ^d and space n ^e .