On (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-relaxed strong edge-coloring of graphs

In this paper, we introduce a new relaxation of strong edge-coloring. Let G be a graph. For two nonnegative integers s and t, an (s, t)-relaxed strong k-edge-coloring is an assignment of k colors to the edges of G, such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. The (s, t)-relaxed strong chromatic index, denoted by \({\chi '}_{(s,t)}(G)\), is the minimum number k of an (s, t)-relaxed strong k-edge-coloring admitted by G. This paper studies the (s, t)-relaxed strong edge-coloring of graphs, especially trees. For a tree T, the tight upper bounds for \({\chi '}_{(s,0)}(T)\) and \({\chi '}_{(0,t)}(T)\) are given. And the (1, 1)-relaxed strong chromatic index of an infinite regular tree is determined. Further results on \({\chi '}_{(1,0)}(T)\) are also presented.     

A note on (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-relaxed <Emphasis Type="Italic">L</Emphasis>(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).     

Approximating <Emphasis Type="Italic">k</Emphasis>-generalized connectivity via collapsing HSTs

An instance of the k -generalized connectivity problem consists of an undirected graph G=(V,E), whose edges are associated with non-negative costs, and a collection \({\mathcal{D}}=\{(S_{1},T_{1}),\ldots,(S_{d},T_{d})\}\) of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph ??G connects a demand (S _i ,T _i ) when it contains a path with one endpoint in S _i and the other in T _i . Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in \({\mathcal{D}}\).Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiple-choice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log?² nlog?² d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k -subgraph.In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n ^2/3?polylog(n,k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.     

New algorithms for online rectangle filling with <Emphasis Type="BoldItalic">k</Emphasis>-lookahead

We study the online rectangle filling problem which arises in channel aware scheduling of wireless networks, and present deterministic and randomized results for algorithms that are allowed a k-lookahead for k≥2. Our main result is a deterministic min {1.848,1+2/(k−1)}-competitive online algorithm. This is the first algorithm for this problem with a competitive ratio approaching 1 as k approaches +∞. The previous best-known solution for this problem has a competitive ratio of 2 for any k≥2. We also present a randomized online algorithm with a competitive ratio of 1+1/(k+1). Our final result is a closely matching lower bound (also proved in this paper) of $1+1/(\sqrt{k+2}+\sqrt{k+1})^{2}>1+1/(4(k+2))$1+1/(\sqrt{k+2}+\sqrt{k+1})^{2}>1+1/(4(k+2)) on the competitive ratio of any randomized online algorithm against an oblivious adversary. These are the first known results for randomized algorithms for this problem.     

On the online multi-agent O–D <Emphasis Type="Italic">k</Emphasis>-Canadian Traveler Problem

In this article, we present new results on the online multi-agent O–D k-Canadian Traveler Problem, in which there are multiple agents and an input graph with a given source node O and a destination node D together with edge costs such that at most k edges are blocked. The blocked edges are not known a priori and are not recoverable. All of the agents are initially located at O. The objective is to find an online strategy such that at least one of the agents finds a route from the initial location O to a given destination D with minimum total cost. We focus on the case where communication among the agents is limited in the sense that some travelers can both send and receive information while the others can only receive information. We formalize the definition of agents’ intelligence by specifying three levels. We introduce two online strategies which utilize higher levels of agents’ intelligence to provide updated lower bounds to this problem. We show that one of our strategies is optimal in both cases with complete and limited communication in the special case of O–D edge-disjoint graphs and highest level of agents’ intelligence.     

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