Computer Science

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k , and the objective is to decide whether G contains a minimal vertex cover of size at least k . Motivated by the kernelization of MMVC with parameter k , our main contribution is to introduce a simple general framework to obtain lower bounds on the degrees of a certain type of polynomial kernels for vertex optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H -free graphs for several graphs H , such as the bull, the paw, or the complete graphs, by making use of the Erd?s-Hajnal property in order to find an appropriate decomposition. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP?�coNP/poly . This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC.

For a matrix M and a positive integer r , the rank r rigidity of M is the smallest number of entries of M which one must change to make its rank at most r . There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: ??For any d>1 , and over any field F , the N?N Walsh-Hadamard transform has a depth- d linear circuit of size O(d??N 1+0.96/d ) . This circumvents a known lower bound of Ω(d??N 1+1/d ) for circuits with bounded coefficients over C by Pudlák (2000), by using coefficients of magnitude polynomial in N . Our construction also generalizes to linear transformations given by a Kronecker power of any fixed 2?2 matrix. ??The N?N Walsh-Hadamard transform has a linear circuit of size ??1.81+o(1))N log 2 N , improving on the bound of ??.88N log 2 N which one obtains from the standard fast Walsh-Hadamard transform. ??A new rigidity upper bound, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant's approach: ??for any field F and any function f:{0,1 } n ?�F , the matrix V f ??F 2 n ? 2 n given by, for any x,y?�{0,1 } n , V f [x,y]=f(x?�y) , and ??for any field F and any fixed-size matrices M 1 ,?? M n ??F q?q , the Kronecker product M 1 ??M 2 ?�⋯??M n . This generalizes recent results on non-rigidity, using a simpler approach which avoids needing the polynomial method.

This report contains the description of two novel job shop scheduling benchmarks that resemble instances of real scheduling problem as they appear in industry. In particular, the aim was to provide large-scale benchmarks (up to 1 million operations) to test the state-of-the-art scheduling solutions on problems that are closer to what occurs in a real industrial context. The first benchmark is an extension of the well known Taillard benchmark (1992), while the second is a collection of scheduling instances with a known-optimum solution.

The aim of this research is a practical method to draw cable plans of complex machines. Such plans consist of electronic components and cables connecting specific ports of the components. Since the machines are configured for each client individually, cable plans need to be drawn automatically. The drawings must be well readable so that technicians can use them to debug the machines. In order to model plug sockets, we introduce port groups; within a group, ports can change their position (which we use to improve the aesthetics of the layout), but together the ports of a group must form a contiguous block. We approach the problem of drawing such cable plans by extending the well-known Sugiyama framework such that it incorporates ports and port groups. Since the framework assumes directed graphs, we propose several ways to orient the edges of the given undirected graph. We compare these methods experimentally, both on real-world data and synthetic data that carefully simulates real-world data. We measure the aesthetics of the resulting drawings by counting bends and crossings. Using these metrics, we compare our approach to Kieler [JVLC 2014], a library for drawing graphs in the presence of port constraints.

Studying on networked systems, in which a communication between nodes is functional if their distance under a given metric is lower than a pre-defined threshold, has received significant attention recently. In this work, we propose a metric to measure network resilience on guaranteeing the pre-defined performance constraint. This metric is investigated under an optimization problem, namely \textbf{Length-bounded Paths Interdiction in Continuous Domain} (cLPI), which aims to identify a minimum set of nodes whose changes cause routing paths between nodes become undesirable for the network service. We show the problem is NP-hard and propose a framework by designing two oracles, \textit{Threshold Blocking} (TB) and \textit{Critical Path Listing} (CPL), which communicate back and forth to construct a feasible solution to cLPI with theoretical bicriteria approximation guarantees. Based on this framework, we propose two solutions for each oracle. Each combination of one solution to \tb and one solution to \cpl gives us a solution to cLPI. The bicriteria guarantee of our algorithms allows us to control the solutions's trade-off between the returned size and the performance accuracy. New insights into the advantages of each solution are further discussed via experimental analysis.

A run in a string is a maximal periodic substring. For example, the string bananatree contains the runs anana=(an ) 3/2 and ee= e 2 . There are less than n runs in any length- n string, and computing all runs for a string over a linearly-sortable alphabet takes O(n) time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an O(nα(n)) time algorithm (where α(n) is the extremely slowly growing inverse Ackermann function). We show how to achieve O(n) time by exploiting combinatorial properties of the Lyndon array, thus proving Kosolobov's conjecture.

A directed graph G=(V,E) is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph G are its maximal twinless strongly connected subgraphs. These concepts have several diverse applications, such as the design of telecommunication networks and the structural stability of buildings. A vertex v∈V is a twinless strong articulation point of G if the deletion of v increases the number of TSCCs of G . Here, we present the first linear-time algorithm that finds all the twinless strong articulation points of a directed graph. We show that the computation of twinless strong articulation points reduces to the following problem in undirected graphs, which may be of independent interest: Given a 2 -vertex-connected (biconnected) undirected graph H , find all vertices v that belong to a vertex-edge cut-pair, i.e., for which there exists an edge e such that H∖{v,e} is not connected. We develop a linear-time algorithm that not only finds all such vertices v , but also computes the number of edges e such that H∖{v,e} is not connected. This also implies that for each twinless strong articulation point v which is not a strong articulation point in a strongly connected digraph G , we can compute the number of TSCCs in G∖v . We note that the problem of computing all vertices that belong to a vertex-edge cut-pair can be solved in linear-time by exploiting the structure of 3 -vertex-connected (triconnected) components of H , represented by an SPQR tree of H . Our approach, however, is conceptually simple, and thus likely to be more amenable to practical implementations.

A colouring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that c(u)≠c(v) for every two adjacent vertices u and v of G . The {\sc List k -Colouring} problem is to decide whether a graph G=(V,E) with a list L(u)⊆{1,…,k} for each u∈V has a colouring c such that c(u)∈L(u) for every u∈V . Let P t be the path on t vertices and let K 1 1,s be the graph obtained from the (s+1) -vertex star K 1,s by subdividing each of its edges exactly once.Recently, Chudnovsky, Spirkl and Zhong (DM 2020) proved that List 3 -Colouring is polynomial-time solvable for ( K 1 1,s , P t ) -free graphs for every t≥1 and s≥1 . We generalize their result to List k -Colouring for every k≥1 . Our result also generalizes the known result that for every k≥1 and s≥0 , List k -Colouring is polynomial-time solvable for (s P 1 + P 5 ) -free graphs, which was proven for s=0 by Hoàng, Kamiński, Lozin, Sawada, and Shu (Algorithmica 2010) and for every s≥1 by Couturier, Golovach, Kratsch and Paulusma (Algorithmica 2015). We show our result by proving boundedness of an underlying width parameter. Namely, we show that for every k≥1 , s≥1 , t≥1 , the class of ( K k , K 1 1,s , P t ) -free graphs has bounded mim-width and that a corresponding branch decomposition is "quickly computable" for these graphs.

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t) . We desire local access algorithms supporting position(G,s,t) queries, which return the position of a random walk from some start vertex s at time t , where the joint distribution of returned positions is 1/poly(n) close to the uniform distribution over such walks in ??1 distance. We first give an algorithm for local access to walks on undirected regular graphs with O ? ( 1 1?��?n ??????) runtime per query, where λ is the second-largest eigenvalue in absolute value. Since random d -regular graphs are expanders with high probability, this gives an O ? ( n ??????) algorithm for G(n,d) , which improves on the naive method for small numbers of queries. We then prove that no that algorithm with sub-constant error given probe access to random d -regular graphs can have runtime better than Ω( n ??????/log(n)) per query in expectation, obtaining a nearly matching lower bound. We further show an Ω( n 1/4 ) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We extend our results to graphs constructed using the tensor product (giving local access to walks on degree n ϵ graphs for any ϵ??0,1] ) and Cartesian product.

Linial's famous color reduction algorithm reduces a given m -coloring of a graph with maximum degree Δ to a O( Δ 2 logm) -coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m -coloring in a directed graph of maximum outdegree β , if every node has a list of size Ω( β 2 (logβ+loglogm+loglog|C|)) from a color space C then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial's color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local (deg+1) -list coloring algorithm from Barenboim et al. [PODC'18] by slightly reducing the runtime to O( ΔlogΔ − − − − − − √ )+ log ∗ n and significantly reducing the message size (from huge to roughly Δ ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. [FOCS'16].