Comb Optim Research Articles

A Recursive Algorithm for Generalized Constraint SCS Problem

In this paper a constraint shortest common supersequence problem, i.e. the string including constraint shortest common supersequence problem was considered. Shortest common supersequence is known as classical problem in the area of strings and it is classified as NP-hard problem. For two input DNA sequences X and Y of length m and n respectively and the constraint DNA sequence P of length r, and the problem was to compute the shortest common supersequence of string X and Y including P as substring and the length of SCS must be minimum. Chen and Chao (J Comb Optim 21:383–392, 2011) proposed the same problem but with constraint as longest common subsequence and its solution in 2011. A recursive procedure for STR-IC-SCS problem was proposed.

Solution of Bharathi–Kempe–Salek conjecture for influence maximization on arborescence

Influence maximization is an important problem in social networks. Bharathi et al. (Competitive influence maximization in social networks, pp 306---311, 2007) conjectured that this problem is $${{\mathcal {N}}}{{\mathcal {P}}}$$NP-hard on arborescence directed into a root. In this short note, we show that the conjecture is true for the independent cascade (IC) model, which is the most studied model in the literature specifying how each node influences other nodes. Therefore, assuming $${\mathcal {P}}\ne {{\mathcal {N}}}{{\mathcal {P}}}$$PźNP, there exists no polynomial-time algorithm for the influence maximization problem under the IC model on arborescence directed into a root. On the other hand, Wang et al. (J Comb Optim, doi:10.1007/s10878-016-9991-1, 2016) have shown that there exists polynomial-time algorithm for this problem under the linear threshold (LT) model. Hence, this pair of results is of theoretical interest since this is the first time to give a theoretical difference on computational complexity between the IC and LT models. We believe it may motivate further research for influence maximization on arborescence and other special graphs.

A note on (s, t)-relaxed L(2, 1)-labeling of graphs

Let \(G=(V, E)\) be a graph. For two vertices u and v in G, we denote \(d_G(u, v)\) the distance between u and v. A vertex v is called an i-neighbor of u if \(d_G(u,v)=i\). Let s, t and k be nonnegative integers. An (s, t)-relaxed k-L(2, 1)-labeling of a graph G is an assignment of labels from \(\{0, 1, \ldots , k\}\) to the vertices of G if the following three conditions are met: (1) adjacent vertices get different labels; (2) for any vertex u of G, there are at most s 1-neighbors of u receiving labels from \(\{f(u)-1,f(u)+1\}\); (3) for any vertex u of G, the number of 2-neighbors of u assigned the label f(u) is at most t. The (s, t)-relaxed L(2, 1)-labeling number \(\lambda _{2,1}^{s,t}(G)\) of G is the minimum k such that G admits an (s, t)-relaxed k-L(2, 1)-labeling. In this article, we refute Conjecture 4 and Conjecture 5 stated in (Lin in J Comb Optim. doi:10.1007/s10878-014-9746-9, 2013).

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 $$O(mn+n^2 \log n)$$O(mn+n2logn) 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 $$O(m+n \log n)$$O(m+nlogn) 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 $$O(m+n \log n)$$O(m+nlogn) time.

Improved algorithmic results for unsplittable stable allocation problems

The allocation problem is a many-to-many generalization of the well-known marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is stable based on a set of underlying preference lists submitted by the jobs and machines. Building on the initial work of Dean et al. (The marriage problem, 2006), we study a natural unsplittable variant of this problem, where each assigned job must be fully assigned to a single machine. Such bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the allocation problem (McDermid and Manlove in J Comb Optim 19(3): 279---303, 2010). Our main result is to show that under an alternative model of stability, the allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed in McDermid and Manlove (J Comb Optim 19(3): 279---303, McDermid and Manlove 2010), we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a allocation problem to produce relaxed unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.

Signed Roman $$k$$ k -Domination in Graphs

Let $$k\ge 1$$k?1 be an integer, and let $$G$$G be a finite and simple graph with vertex set $$V(G)$$V(G). A signed Roman $$k$$k-dominating function (SRkDF) on a graph $$G$$G is a function $$f:V(G)\rightarrow \{-1,1,2\}$$f:V(G)?{-1,1,2} satisfying the conditions that (i) $$\sum _{x\in N[v]}f(x)\ge k$$?x?N[v]f(x)?k for each vertex $$v\in V(G)$$v?V(G), where $$N[v]$$N[v] is the closed neighborhood of $$v$$v, and (ii) every vertex $$u$$u for which $$f(u)=-1$$f(u)=-1 is adjacent to at least one vertex $$v$$v for which $$f(v)=2$$f(v)=2. The weight of an SRkDF $$f$$f is $$w(f)=\sum _{v\in V(G)}f(v)$$w(f)=?v?V(G)f(v). The signed Roman $$k$$k-domination number $$\gamma _{sR}^k(G)$$?sRk(G) of $$G$$G is the minimum weight of an SRkDF on $$G$$G. In this paper we initiate the study of the signed Roman $$k$$k-domination number of graphs, and we present different bounds on $$\gamma _{sR}^k(G)$$?sRk(G). In addition, we determine the signed Roman $$k$$k-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman domination number $$\gamma _{sR}(G)=\gamma _{sR}^1(G)$$?sR(G)=?sR1(G), introduced and investigated by Ahangar et al. (J Comb Optim doi:10.1007/s10878-012-9500-0, 2014).

Revisiting wireless network jamming by SIR-based considerations and multiband robust optimization

We revisit the mathematical models for wireless network jamming introduced by Commander et al. (J Comb Optim 14:481–498 2007, Optim Lett 2(1):53–70 2008): we first point out the strong connections with classical wireless network design and then we propose a new model based on the explicit use of signal-to-interference quantities. Moreover, to address the uncertain nature of the jamming problem and tackle the peculiar right-hand-side uncertainty of the corresponding model, we propose an original robust cutting-plane algorithm drawing inspiration from multiband robust optimization. Finally, we assess the performance of the proposed cutting plane algorithm by experiments on realistic network instances.

Novel GA for metropolitan stations of Indian railways when modelled as a TSP

In this paper, seven cities that have a direct connection link by Indian railways are modeled as a travelling salesman problem. Then genetic algorithm (GA) is used to solve it by considering three different objective functions, namely: distance, cost and time. For the implementation of GA, the fourth variation of order crossover (OX4) as proposed in Deep and Mebrahtu (Int J Comb Optim Probl Inform 2(3):1–23, 2011a) with inversion mutation and inverted displacement mutations are used. These are programmed in C++ and implemented on the distance, cost and time data obtained from the Indian railways. The minimum and maximum distances of travel, costs of travel and time taken to cover the stations are evaluated. According to the analysis of results that is based on numerical experimentations the sequence of choosing stations really matters. This is observed by the big difference between the minimum and maximum distance, cost and time of travel evaluated. Especially the difference between the minimum and maximum results of distance travelled and time taken to cover the tours is almost twice.

Some extremal properties of the multiplicatively weighted Harary index of a graph

Let $$G=(V_G, E_G)$$G=(VG,EG) be a simple connected graph. The multiplicatively weighted Harary index of $$G$$G is defined as $$H_M(G)=\sum _{\{u,v\}\subseteq V_G}\delta _G(u)\delta _G(v)\frac{1}{d_G(u,v)},$$HM(G)=?{u,v}⊆VG?G(u)?G(v)1dG(u,v), where $$\delta _G(u)$$?G(u) is the vertex degree of $$u$$u and $$d_G(u,v)$$dG(u,v) is the distance between $$u$$u and $$v$$v in $$G.$$G. This novel invariant is in fact the modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. Deng et al. (J Comb Optim 2014, doi:10.1007/s10878-013-9698-5) determined the extremal values on $$H_M$$HM of graphs among $$n$$n-vertex trees (resp. unicyclic graphs). In this paper, as a continuance of it, the monotonicity of $$H_M(G)$$HM(G) under some graph transformations were studied. Using these nice mathematical properties, the extremal graphs among $$n$$n-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. are characterized, respectively. Some sharp upper bounds on the multiplicatively weighted Harary index of trees with given parameters are determined.

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85---106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based $$\alpha $$?-approximation for the original covering problem can be leveraged to obtain an $$\left( 1-e^{-1/\alpha }\right) ^{-1}$$1-e-1/?-1-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best $$2$$2-approximation ratio over the previously best $$2.375$$2.375 by Li et al. (Theoret Comput Sci 476:109---117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best $$1.5148$$1.5148-approximation ratio over the previously best $$1.853$$1.853 by Xu and Xu (J Comb Optim 17:424---436, 2008) for the FLPLP, respectively.

$$L(2,1)$$ L ( 2 , 1 ) -labeling for brick product graphs

Let $$G=(V, E)$$G=(V,E) be a graph. Denote $$d_G(u, v)$$dG(u,v) the distance between two vertices $$u$$u and $$v$$v in $$G$$G. An $$L(2, 1)$$L(2,1)-labeling of $$G$$G is a function $$f: V \rightarrow \{0,1,\cdots \}$$f:V?{0,1,?} such that for any two vertices $$u$$u and $$v$$v, $$|f(u)-f(v)| \ge 2$$|f(u)-f(v)|?2 if $$d_G(u, v) = 1$$dG(u,v)=1 and $$|f(u)-f(v)| \ge 1$$|f(u)-f(v)|?1 if $$d_G(u, v) = 2$$dG(u,v)=2. The span of $$f$$f is the difference between the largest and the smallest number in $$f(V)$$f(V). The $$\lambda $$?-number of $$G$$G, denoted $$\lambda (G)$$?(G), is the minimum span over all $$L(2,1 )$$L(2,1)-labelings of $$G$$G. In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716---736, 2013) in the case when (i) $$\ell $$l is even, or (ii) $$\ell \ge 5$$l?5 is odd and $$0 \le r \le 8$$0≤r≤8.

Dual power assignment optimization and fault tolerance in WSNs

Because of limited battery equipped on each sensor, power consumption is one of the crucial issues in wireless sensor networks (WSNs). It therefore has been the focus of many researchers. An important problem concerning power consumption is to minimize the number of maximum-power nodes while maintaining a desired network topology. As fault tolerance is vitally important in practice, it is desirable that the constructed network topology is $$k$$k-edge-connected or $$k$$k-connected. In this paper, we study the dual power assignment problem for $$k$$k-edge connectivity $$(kEDP)$$(kEDP) and biconnectivity in WSNs. While other studies consider only the special case $$k=2$$k=2, our goal is to address the general problem. In addition to showing the APX-completeness of biconnectivity problem in the metric model, we also prove the NP-completeness of the $$kEDP$$kEDP problem in the geometric case and provide a 2-approximation algorithm using linear programming techniques. To the best of our knowledge, this approximation ratio is currently the best one. We also introduce a heuristic whose performance is better compared with an approximation algorithm in Wang et al. (J Comb Optim 19:174---183, 2010).

L( $$d$$ d ,1)-labelings of the edge-path-replacement by factorization of graphs

For an integer $$d \ge 2$$d?2, an $$L(d$$L(d,1)-labeling of a graph $$G$$G is a function $$f$$f from its vertex set to the non-negative integers such that $${\vert }f(x) - f(y){\vert } \ge d$$|f(x)?f(y)|?d if vertices $$x$$x and $$y$$y are adjacent, and $${\vert }f(x) - f(y){\vert } \ge $$|f(x)?f(y)|? 1 if $$x$$x and $$y$$y are at distance two. The minimum span over all the L($$d$$d,1)-labelings of $$G$$G is denoted by $$\lambda _{d}(G)$$?d(G). For a given integer $$k \ge 2$$k?2, the edge-path-replacement of$$G$$G or $$G(P_{k})$$G(Pk) is the graph obtained from $$G$$G by replacing each edge with a path $$P_{k}$$Pk on $$k$$k vertices. We show that the edges of $$G$$G can be colored with $$\lceil \varDelta (G)/2\rceil $$?Δ(G)/2? colors so that each monochromatic subgraph has maximum degree at most 2 and use this fact to establish general upper bounds on $$\lambda _{d}(G(P_{k}))$$?d(G(Pk)) for $$k \ge 4$$k?4. As a corollary, we settle the following conjecture by Lu (J Comb Optim, 2012): for any graph $$G$$G with $$\varDelta (G) \ge $$Δ(G)? 2, $$\lambda _{2}(G(P_{4})) \le \varDelta (G)$$?2(G(P4))≤Δ(G) + 2. Moreover, $$\lambda _{2}(G(P_{4})) = \varDelta (G) + 1$$?2(G(P4))=Δ(G)+1 when $$\varDelta (G)$$Δ(G) is even and different from 2. We also show that the class of graphs $$G(P_{k})$$G(Pk) with $$k \ge $$k? 4 satisfies a conjecture by Havet and Yu (2008 Discrete Math 308:498---513) in the related area of ($$d,1$$d,1)-total labeling of graphs.

Efficient polynomial-time algorithms for the constrained LCS problem with strings exclusion

In this paper, we revisit a recent variant of the longest common subsequence (LCS) problem, the string-excluding constrained LCS (STR-EC-LCS) problem, which was first addressed by Chen and Chao (J Comb Optim 21(3):383---392, 2011). Given two sequences $$X$$ and $$Y$$ of lengths $$m$$ and $$n,$$ respectively, and a constraint string $$P$$ of length $$r,$$ we are to find a common subsequence $$Z$$ of $$X$$ and $$Y$$ which excludes $$P$$ as a substring and the length of $$Z$$ is maximized. In fact, this problem cannot be correctly solved by the previously proposed algorithm. Thus, we give a correct algorithm with $$O(mnr)$$ time to solve it. Then, we revisit the STR-EC-LCS problem with multiple constraints $$\{ P_1, P_2, \ldots , P_k \}.$$ We propose a polynomial-time algorithm which runs in $$O(mnR)$$ time, where $$R = \sum _{i=1}^{k} |P_i|,$$ and thus it overthrows the previous claim of NP-hardness.

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤ r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K 1,n for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤ rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 − 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1–18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in Zhao et al. (J Comb Optim 2:71–109, 1998). Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP instances where the data matrices have large automorphism groups, these bounds can be computed more efficiently, as was shown in Klerk and Sotirov (Math Program A, 122(2), 225–246, 2010). Continuing in the same vein, we show how one may obtain stronger bounds for QAP instances where one of the data matrices has a transitive automorphism group. To illustrate our approach, we compute improved lower bounds for several instances from the QAP library QAPLIB.

Paired-Domination in Subdivided Star-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γ pr(G), is the minimum cardinality of a paired-dominating set of G. In [Dorbec P, Gravier S, Henning MA, J Comb Optim 14(1):1–7, 2007], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced subdivided stars, depending on the size of the star.