Finding an anti-risk path between two nodes in undirected graphs

Given a weighted graph G=(V,E) with a source s and a destination t, a traveler has to go from s to t. However, some of the edges may be blocked at certain times, and the traveler only observes that upon reaching an adjacent site of the blocked edge. Let ℘={P _G (s,t)} be the set of all paths from s to t. The risk of a path is defined as the longest travel under the assumption that any edge of the path may be blocked. The paper will propose the Anti-risk Path Problem of finding a path P _G (s,t) in ℘ such that it has minimum risk. We will show that this problem can be solved in O(mn+n ²log n) time suppose that at most one edge may be blocked, where n and m denote the number of vertices and edges in G, respectively. This research is supported by NSF of China under Grants 70525004, 60736027, 70121001 and Postdoctoral Science Foundation of China under Grant 20060401003.     

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.     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

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).     

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.     

Finding a contra-risk path between two nodes in undirected graphs

Given an undirected graph with a source node s and a sink node t. The anti-risk path problem is defined as the problem of finding a path between node s to node t with the least risk under the assumption that at most one edge of each path may be blocked. Xiao et al. (J Comb Optim 17:235–246, 2009) defined the problem and presented an \(O(mn+n^2 \log n)\) time algorithm to find an anti-risk path, where n and m are the number of nodes and edges, respectively. Recently, Mahadeokar and Saxena (J Comb Optim 27:798–807, 2014) solved the problem in \(O(m+n \log n)\) time. In this paper, first, a new version of the anti-risk path (called contra-risk path) is defined, which is more effective than an anti-risk path in many networks. Then, an algorithm to find a contra-risk path is presented, which runs in \(O(m+n \log n)\) time.     

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ?-colourings of G is connected and has diameter O(|V|²), for all ?≥k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|²).     

Parity and strong parity edge-colorings of graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. A parity edge-coloring (respectively, strong parity edge-coloring) is an edge-coloring in which there is no nontrivial parity path (respectively, open parity walk). The parity edge-chromatic number p(G) (respectively, strong parity edge-chromatic number $\widehat{p}(G)$ ) is the least number of colors in a parity edge-coloring (respectively, strong parity edge-coloring) of G. Notice that $\widehat{p}(G) \ge p(G) \ge \chi'(G) \ge \Delta(G)$ for any graph G. In this paper, we determine $\widehat{p}(G)$ and p(G) for some complete bipartite graphs and some products of graphs. For instance, we determine $\widehat{p}(K_{m,n})$ and p(K _m,n ) for m≤n with n≡0,?1,?2 (mod 2^?lg?m?).     

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.     

Minimum Cost Edge Subset Covering Exactly k Vertices of a Graph

Given a graph G with nonnegative edge costs and an integer k, we consider the problem of finding an edge subset S of minimum total cost with respect to the constraint that S covers exactly k vertices of G. An O(n ³) algorithm is presented where n is the order of G. It is based on the author's previous paper dealing with a similar problem asking S to cover at least k vertices.     

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.     

A dynamic programming approach of finding an optimal broadcast schedule in minimizing total flow time

We study the problem of (off-line) broadcast scheduling in minimizing total flow time and propose a dynamic programming approach to compute an optimal broadcast schedule. Suppose the broadcast server has k pages and the last page request arrives at time n. The optimal schedule can be computed in O(k³(n+k)^k−1) time for the case that the server has a single broadcast channel. For m channels case, i.e., the server can broadcast m different pages at a time where m < k, the optimal schedule can be computed in O(n^k−m) time when k and m are constants. Note that this broadcast scheduling problem is NP-hard when k is a variable and will take O(n^k−m+1) time when k is fixed and m ≥ 1 with the straightforward implementation of the dynamic programming approach. The preliminary version of this paper appeared in Proceedings of the 11th Annual International Computing and Combinatorics Conference as “Off-line Algorithms for Minimizing the Total Flow Time in Broadcast Scheduling”.     

Two constructions of new error-correcting pooling designs from orthogonal spaces over a finite field of characteristic 2

In this paper, we construct two classes of t×n,s ^e -disjunct matrix with subspaces in orthogonal space \mathbbF_q⁽²ⁿ⁺¹⁾\mathbb{F}_{q}^{(2\nu+1)} of characteristic 2 and exhibit their disjunct properties. We also prove that the test efficiency t/n of constructions II is smaller than that of D’yachkov et al. (J. Comput. Biol. 12:1129–1136, 2005).     

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.     

Multiple L(j,1)-labeling of the triangular lattice

Let n,j,k be nonnegative integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a?b|≥j if uv∈E(G), and |a?b|≥k if u and v are distance two apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G. Let n,j,k and m be nonnegative integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m?1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a?b|,m?|a?b|}≥j if uv∈E(G), and min{|a?b|,m?|a?b|}≥k if u and v are distance two apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G. This paper provides upper and lower bounds for the n-fold L(j,1)-labeling number and the n-fold circular L(j,1)-labeling number of the triangular lattice and determines the n-fold L(2,1)-labeling number and n-fold circular L(2,1)-labeling number of the triangular lattice for n≥3.     

Finding a Length-Constrained Maximum-Density Path in a Tree

Let T = (V,E,w) be an undirected and weighted tree with node set V and edge set E, where w(e) is an edge weight function for e E. The density of a path, say e₁, e₂,..., e_k, is defined as ^k_{i = 1} w(e_i)/k. The length of a path is the number of its edges. Given a tree with n edges and a lower bound L where 1 L n, this paper presents two efficient algorithms for finding a maximum-density path of length at least L in O(nL) time. One of them is further modified to solve some special cases such as full m-ary trees in O(n) time.     

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.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.