Improved approximations for buy-at-bul and shallow-light $$$$-Steiner trees and $$(,2)$$(,2)-subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light \(k\)-Steiner tree problem (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). The goal is to find a minimum total cost \(r\)-rooted Steiner tree \(H\) containing at least \(k\) terminals, where the cost of each edge \(e\) is \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost \(2\)-edge-connected subgraph with at least \(k\) vertices, which is introduced as the \((k,2)\)-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).     

Distributed wireless link scheduling in the SINR model

We present an approximation algorithm for wireless link scheduling under the physical SINR interference model. In the link scheduling problem, it is given a set of \(n\) links in a metric space, each of which is a sender–receiver pair, and the objective is to schedule the links using the minimum amount of time. We focus on a variant of this fundamental problem where the power is fixed, i.e., the power assignment of links is given as part of the input. Specifically, we consider an important category of power assignments called length-monotone sublinear power assignment, which includes the widely studied uniform, mean and linear power assignments. We present a distributed algorithm that can schedule all links in \(O(\log \varDelta (I_{max}+\log ^3n))\) rounds with high probability, where \(\varDelta \) is the ratio between the longest link and the shortest link and \(I_{max}\) is the maximum nearly-equilength class affectance of the link set. It is shown that the proposed algorithm is \(O(\log \varDelta )\) approximate to the optimal schedule in dense networks with \(I_{max}\in \varOmega (\log ^3n)\). To the best of our knowledge, our algorithm is the first distributed one whose approximation ratio is independent of the network size \(n\). Our result also shows that the \(\varOmega (\log n)\) lower bound (Halldórsson and Mitra in: ICALP, 2011) on the approximation ratio does not hold for link sets with \(\log \varDelta \in o(\log n)\).     

Dynamic matchings in left vertex weighted convex bipartite graphs

A left vertex weighted convex bipartite graph (LWCBG) is a bipartite graph \(G=(X,Y,E)\) in which the neighbors of each \(x\in X\) form an interval in \(Y\) where \(Y\) is linearly ordered, and each \(x\in X\) has an associated weight. This paper considers the problem of maintaining a maximum weight matching in a dynamic LWCBG. The graph is subject to the updates of vertex and edge insertions and deletions. Our dynamic algorithms maintain the update operations in \(O(\log ^2{|V|})\) amortized time per update, obtain the matching status of a vertex (whether it is matched) in constant worst-case time, and find the pair of a matched vertex (with which it is matched) in worst-case \(O(k)\) time, where \(k\) is not greater than the cardinality of the maximum weight matching. That achieves the same time bound as the best known solution for the problem of the unweighted version.     

An approximation algorithm for maximum weight budgeted connected set cover

This paper studies approximation algorithm for the maximum weight budgeted connected set cover (MWBCSC) problem. Given an element set \(X\), a collection of sets \({\mathcal {S}}\subseteq 2^X\), a weight function \(w\) on \(X\), a cost function \(c\) on \({\mathcal {S}}\), a connected graph \(G_{\mathcal {S}}\) (called communication graph) on vertex set \({\mathcal {S}}\), and a budget \(L\), the MWBCSC problem is to select a subcollection \({\mathcal {S'}}\subseteq {\mathcal {S}}\) such that the cost \(c({\mathcal {S'}})=\sum _{S\in {\mathcal {S'}}}c(S)\le L\), the subgraph of \(G_{\mathcal {S}}\) induced by \({\mathcal {S'}}\) is connected, and the total weight of elements covered by \({\mathcal {S'}}\) (that is \(\sum _{x\in \bigcup _{S\in {\mathcal {S'}}}S}w(x)\)) is maximized. We present a polynomial time algorithm for this problem with a natural communication graph that has performance ratio \(O((\delta +1)\log n)\), where \(\delta \) is the maximum degree of graph \(G_{\mathcal {S}}\) and \(n\) is the number of sets in \({\mathcal {S}}\). In particular, if every set has cost at most \(L/2\), the performance ratio can be improved to \(O(\log n)\).     

Partial degree bounded edge pacing problem for graphs and $$$$-uniform hypergraphs

Given a graph \(G=(V,E)\) and a non-negative integer \(c_u\) for each \(u\in V\), partial degree bounded edge packing problem is to find a subgraph \(G^{\prime }=(V,E^{\prime })\) with maximum \(|E^{\prime }|\) such that for each edge \((u,v)\in E^{\prime }\), either \(deg_{G^{\prime }}(u)\le c_u\) or \(deg_{G^{\prime }}(v)\le c_v\). The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all \(c_u\) being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors \(4\) and \(2\). Then we give a \(\log _2 n\) approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity \(O(n\log n)\). We also consider a generalization of this problem to \(k\)-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.     

The 3-rainbow index and connected dominating sets

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of \(G\) is called a 3-rainbow coloring if for any three vertices in \(G\), there exists a rainbow tree connecting them. The 3-rainbow index \(rx_3(G)\) of \(G\) is defined as the minimum number of colors that are needed in a 3-rainbow coloring of \(G\). This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected 3-way dominating sets and 3-dominating sets. We show that for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (3\le \delta \le 5)\), \(rx_{3}(G)\le \frac{3n}{\delta +1}+4\), and the bound is tight up to an additive constant; whereas for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (\delta \ge 3)\), we get that \(rx_{3}(G)\le \frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))n+5\). In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs.     

Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let \(G\rightarrow (H,H)\) signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote \(\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}\). For any \(\Delta \) and n, we say that G is partition universal for \(\mathcal {H}(\Delta ,n)\) if \(G\rightarrow (H,H)\) for every \(H\in \mathcal {H}(\Delta ,n)\). Let \(G_r(N,p)\) be the random spanning subgraph of the complete r-partite graph \(K_r(N)\) with N vertices in each part, in which each edge of \(K_r(N)\) appears with probability p independently and randomly. We prove that for fixed \(\Delta \ge 2\) there exist constants r, B and C depending only on \(\Delta \) such that if \(N\ge Bn\) and \(p=C(\log N/N)^{1/\Delta }\), then asymptotically almost surely \(G_r(N,p)\) is partition universal for \(\mathcal {H}(\Delta ,n)\).     

Equality in a bound that relates the size and the restrained domination number of a graph

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). The best possible upper bound \(q(n,k)\) is established in Joubert (Discrete Appl Math 161:829–837, 2013) on the size \(m(G)\) of a graph \(G\) with a given order \(n \ge 5\) and restrained domination number \(k \in \{3, \ldots , n-2\}\). We extend this result to include the cases \(k=1,2,n\), and characterize graphs \(G\) of order \(n \ge 1\) and restrained domination number \(k \in \{1,\dots , n-2,n\}\) for which \(m(G)=q(n,k)\).     

Information exchange with collision detection on multiple channels

Information exchange is a fundamental communication primitive in radio networks. We study this problem in multi-channel single-hop networks. In particular, given \(k\) pieces of information, initially stored in \(k\) nodes respectively, the task is to broadcast these information pieces to the entire network via a set of \(\mathcal {F}\) channels. We develop efficient distributed algorithms for this task for the scenario where both the identities and the number \(k\) of the initial information holders are unknown to the participating nodes. Assuming nodes with collision detection, we present an efficient randomized algorithm for unrestricted information exchange, where multiple information items can be combined into a single message. The algorithm disseminates all the information items within \(O(\frac{k}{\mathcal {F}}+\mathcal {F}\log ^2n)\) timeslots with high probability. To the best of our knowledge, this is the first algorithm that breaks the \(\varOmega (k)\) lower bound for unrestricted information exchange if only a single channel is available. This result establishes the superiority of multiple channels for the task of unrestricted information exchange. Moreover, for restricted information exchange, where each message can carry only one information item, we devise a randomized algorithm that completes the task in \(O(k+\frac{\log ^2n}{\mathcal {F}}+\log n)\) timeslots. When \(k\) is large, both algorithms are asymptotically optimal, as they can reach the trivial lower bounds of \(\varOmega (\frac{k}{\mathcal {F}})\) and \(\varOmega (k)\) for unrestricted and restricted information exchange, respectively.     

Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

Constrained pairwise and center-star sequences alignment problems

This paper considers the minimax regret vertex 2-sink location problem in a dynamic path network with positive edge lengths and uniform edge capacity. Let \(P\) be an undirected path graph of \(n\) vertices, and the weight (initial supply) of every vertex is known as an interval. The problem is to find two vertices \(x\) and \(y\) as two sinks on the path such that all the weights can evacuate to \(x\) and \(y\) with minimum regret of evacuation time in case of an emergency for any possible weight distribution. We present an \(O(n^3\log n)\) time algorithm.     

An $$O(n\log n)$$ algorithm for finding edge span of cacti

Let \(G=(V,E)\) be a nonempty graph and \(\xi :E\rightarrow \mathbb {N}\) be a function. In the paper we study the computational complexity of the problem of finding vertex colorings \(c\) of \(G\) such that:

1. (1)
   \(|c(u)-c(v)|\ge \xi (uv)\) for each edge \(uv\in E\);
    
2. (2)
   the edge span of \(c\), i.e. \(\max \{|c(u)-c(v)|:uv\in E\}\), is minimal.
    

We show that the problem is NP-hard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for cycles and bipartite graphs. Next, we use the last two results to construct an algorithm that solves the problem for a given cactus \(G\) in \(O(n\log n)\) time, where \(n\) is the number of vertices of \(G\).

     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).     

A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let \(G=(V,E)\) be a graph and \(\phi \) be a total \(k\)-coloring of \(G\) using the color set \(\{1,\ldots , k\}\). Let \(\sum _\phi (u)\) denote the sum of the color of the vertex \(u\) and the colors of all incident edges of \(u\). A \(k\)-neighbor sum distinguishing total coloring of \(G\) is a total \(k\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(\sum _\phi (u)\ne \sum _\phi (v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak first introduced this coloring and conjectured that \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph \(G\). In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with \(\Delta (G)\ge 7\). Moreover, we also show that \(\chi _{nsd}^{''}(G)\le \Delta (G)+2\) for planar graphs without intersecting triangles with \(\Delta (G) \ge 9\). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.     

Approximation algorithms for minimum weight partial connected set cover problem

In the Minimum Weight Partial Connected Set Cover problem, we are given a finite ground set \(U\), an integer \(q\le |U|\), a collection \(\mathcal {E}\) of subsets of \(U\), and a connected graph \(G_{\mathcal {E}}\) on vertex set \(\mathcal {E}\), the goal is to find a minimum weight subcollection of \(\mathcal {E}\) which covers at least \(q\) elements of \(U\) and induces a connected subgraph in \(G_{\mathcal {E}}\). In this paper, we derive a “partial cover property” for the greedy solution of the Minimum Weight Set Cover problem, based on which we present (a) for the weighted version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are adjacent in \(G_{\mathcal {E}}\) (the Minimum Weight Partial Connected Vertex Cover problem falls into this range), an approximation algorithm with performance ratio \(\rho (1+H(\gamma ))+o(1)\), and (b) for the cardinality version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are at most \(d\)-hops away from each other (the Minimum Partial Connected \(k\)-Hop Dominating Set problem falls into this range), an approximation algorithm with performance ratio \(2(1+dH(\gamma ))+o(1)\), where \(\gamma =\max \{|X|:X\in \mathcal {E}\}\), \(H(\cdot )\) is the Harmonic number, and \(\rho \) is the performance ratio for the Minimum Quota Node-Weighted Steiner Tree problem.     

Cacti with the smallest,second smallest,and third smallest Gutman index

The Gutman index (also known as Schultz index of the second kind) of a graph \(G\) is defined as \(Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)\). A graph \(G\) is called a cactus if each block of \(G\) is either an edge or a cycle. Denote by \(\mathcal {C}(n, k)\) the set of connected cacti possessing \(n\) vertices and \(k\) cycles. In this paper, we give the first three smallest Gutman indices among graphs in \(\mathcal {C}(n, k)\), the corresponding extremal graphs are characterized as well.     

Signed mixed Roman domination numbers in graphs

Let \(G = (V;E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A signed mixed Roman dominating function (SMRDF) of \(G\) is a function \(f: V\cup E\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{y\in N_m[x]}f(y)\ge 1\) for each \(x\in V\cup E\), where \(N_m[x]\) is the set, called mixed closed neighborhood of \(x\), consists of \(x\) and the elements of \(V\cup E\) adjacent or incident to \(x\) (ii) every element \(x\in V\cup E\) for which \(f(x) = -1\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y) = 2\). The weight of a SMRDF \(f\) is \(\omega (f)=\sum _{x\in V\cup E}f(x)\). The signed mixed Roman domination number \(\gamma _{sR}^*(G)\) of \(G\) is the minimum weight of a SMRDF of \(G\). In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.     

An FPTAS for generalized absolute 1-center problem in vertex-weighted graphs

Given a vertex-weighted undirected connected graph \(G = (V, E, \ell , \rho )\), where each edge \(e \in E\) has a length \(\ell (e) > 0\) and each vertex \(v \in V\) has a weight \(\rho (v) > 0\), a subset \(T \subseteq V\) of vertices and a set S containing all the points on edges in a subset \(E' \subseteq E\) of edges, the generalized absolute 1-center problem (GA1CP), an extension of the classic vertex-weighted absolute 1-center problem (A1CP), asks to find a point from S such that the longest weighted shortest path distance in G from it to T is minimized. This paper presents a simple FPTAS for GA1CP by traversing the edges in \(E'\) using a positive real number as step size. The FPTAS takes \(O( |E| |V| + |V|^2 \log \log |V| + \frac{1}{\epsilon } |E'| |T| {\mathcal {R}})\) time, where \({\mathcal {R}}\) is an input parameter size of the problem instance, for any given \(\epsilon > 0\). For instances with a small input parameter size \({\mathcal {R}}\), applying the FPTAS with \(\epsilon = \Theta (1)\) to the classic vertex-weighted A1CP can produce a \((1 + \Theta (1))\)-approximation in at most O(|E| |V|) time when the distance matrix is known and \(O(|E| |V| + |V|^2 \log \log |V|)\) time when the distance matrix is unknown, which are smaller than Kariv and Hakimi’s \(O(|E| |V| \log |V|)\)-time algorithm and \(O(|E| |V| \log |V| + |V|^3)\)-time algorithm, respectively.     

$$$$-labeling for brick product graphs

Let \(G=(V, E)\) be a graph. Denote \(d_G(u, v)\) the distance between two vertices \(u\) and \(v\) in \(G\). An \(L(2, 1)\)-labeling of \(G\) is a function \(f: V \rightarrow \{0,1,\cdots \}\) such that for any two vertices \(u\) and \(v\), \(|f(u)-f(v)| \ge 2\) if \(d_G(u, v) = 1\) and \(|f(u)-f(v)| \ge 1\) if \(d_G(u, v) = 2\). The span of \(f\) is the difference between the largest and the smallest number in \(f(V)\). The \(\lambda \)-number of \(G\), denoted \(\lambda (G)\), is the minimum span over all \(L(2,1 )\)-labelings of \(G\). 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) \(\ell \) is even, or (ii) \(\ell \ge 5\) is odd and \(0 \le r \le 8\).     

Parameterized algorithms for min–max 2-cluster editing

For a given graph and an integer t, the Min–Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for \(t<n/4\), where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t. Let \(k=t-n/4\). We show that the problem is polynomial-time solvable when roughly \(k<\sqrt{n/32}\). When \(k\in o(n)\), we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in \(O({2}^{n/r}\cdot n^2)\) time for \(k<n/12\) and in \(O(n^2\cdot 2^{3n/4+k})\) time for \(n/12\le k< n/4\), where \(r=2+\lfloor (n/4-3k-2)/(2k+1) \rfloor \ge 2\).