Computer Science

A strong arc decomposition of a digraph D=(V,A) is a decomposition of its arc set A into two disjoint subsets A 1 and A 2 such that both of the spanning subdigraphs D 1 =(V, A 1 ) and D 2 =(V, A 2 ) are strong. Let T be a digraph with t vertices u 1 ,…, u t and let H 1 ,… H t be digraphs such that H i has vertices u i, j i , 1≤ j i ≤ n i . Then the composition Q=T[ H 1 ,…, H t ] is a digraph with vertex set ∪ t i=1 V( H i )={ u i, j i ∣1≤i≤t,1≤ j i ≤ n i } and arc set ( ∪ t i=1 A( H i ))∪( ∪ u i u p ∈A(T) { u i j i u p q p ∣1≤ j i ≤ n i ,1≤ q p ≤ n p }). We obtain a characterization of digraph compositions Q=T[ H 1 ,… H t ] which have a strong arc decomposition when T is a semicomplete digraph and each H i is an arbitrary digraph. Our characterization generalizes a characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018) on strong arc decompositions of digraph compositions Q=T[ H 1 ,…, H t ] in which T is semicomplete and each H i is arbitrary. Our proofs are constructive and imply the existence of a polynomial algorithm for constructing a \good{} decomposition of a digraph Q=T[ H 1 ,…, H t ] , with T semicomplete, whenever such a decomposition exists.

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r . Let T be a digraph with t vertices u 1 ,…, u t and let H 1 ,… H t be digraphs such that H i has vertices u i, j i , 1≤ j i ≤ n i . Then the composition Q=T[ H 1 ,…, H t ] is a digraph with vertex set { u i, j i ∣1≤i≤t,1≤ j i ≤ n i } and arc set A(Q)= ∪ t i=1 A( H i )∪{ u i j i u p q p ∣ u i u p ∈A(T),1≤ j i ≤ n i ,1≤ q p ≤ n p }. When T is arbitrary, we obtain the following result: every strong digraph composition Q in which n i ≥2 for every 1≤i≤t , has a good pair at every vertex of Q. The condition of n i ≥2 in this result cannot be relaxed. When T is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

We consider the problem of assigning tasks to agents under time conflicts, with applications also to frequency allocations in point-to-point wireless networks. In particular, we are given a set V of n agents, a set E of m tasks, and k different time slots. Each task can be carried out in one of the k predefined time slots, and can be represented by the subset e⊆E of the involved agents. Since each agent cannot participate to more than one task simultaneously, we must find an allocation that assigns non-overlapping tasks to each time slot. Being the number of slots limited by k , in general it is not possible to executed all the possible tasks, and our aim is to determine a solution maximizing the overall social welfare, that is the number of executed tasks. We focus on the restriction of this problem in which the number of time slots is fixed to be k=2 , and each task is performed by exactly two agents, that is |e|=2 . In fact, even under this assumptions, the problem is still challenging, as it remains computationally difficult. We provide parameterized complexity results with respect to several reasonable parameters, showing for the different cases that the problem is fixed-parameter tractable or it is paraNP-hard.

In this text, we prove the existence of an asymptotic growth rate of the number of dominating sets (and variants) on finite rectangular grids, when the dimensions of the grid grow to infinity. Moreover, we provide, for each of the variants, an algorithm which computes the growth rate. We also give bounds on these rates provided by a computer program.

For a complexity function C , the lower and upper C -complexity rates of an infinite word x are C – – (x)= lim inf n→∞ C(x↾n) n , C ¯ ¯ ¯ ¯ (x)= lim sup n→∞ C(x↾n) n respectively. Here x↾n is the prefix of x of length n . We consider the case C= A N , the nondeterministic automatic complexity. If these rates are strictly between 0 and 1/2 , we call them intermediate. Our main result is that words having intermediate A N -rates exist, viz. the infinite Fibonacci and Tribonacci words.

We prove a recent conjecture of Beisegel et al. that for every positive integer k, every graph containing an induced P_k also contains an avoidable P_k. Avoidability generalises the notion of simpliciality best known in the context of chordal graphs. The conjecture was only established for k in {1,2} (Ohtsuki et al. 1976, and Beisegel et al. 2019, respectively). Our result also implies a result of Chvátal et al. 2002, which assumed cycle restrictions. We provide a constructive and elementary proof, relying on a single trick regarding the induction hypothesis. In the line of previous works, we discuss conditions for multiple avoidable paths to exist.

We generalize Axel Thue's familiar definition of overlaps in words, and show that there are no infinite words containing split occurrences of these generalized overlaps. Along the way we prove a useful theorem about repeated disjoint occurrences in words -- an interesting natural variation on the classical de Bruijn sequences.

Ramsey Theory deals with avoiding certain patterns. When constructing an instance that avoids one pattern, it is observed that other patterns emerge. For example, repetition emerges when avoiding arithmetic progression (Van der Waerden numbers), while reflection emerges when avoiding monochromatic solutions of a+b=c (Schur numbers). We exploit observed patterns when coloring a grid while avoiding monochromatic rectangles. Like many problems in Ramsey Theory, this problem has a rapidly growing search space that makes computer search difficult. Steinbach et al. obtained a solution of an 18 by 18 grid with 4 colors by enforcing a rotation symmetry. However, that symmetry is not suitable for 5 colors. In this article, we will encode this problem into propositional logic and enforce so-called internal symmetries, which preserves satisfiability, to guide SAT-solving. We first observe patterns with 2 and 3 colors, among which the "shift pattern" can be easily generalized and efficiently encoded. Using this pattern, we obtain a new solution of the 18 by 18 grid that is non-isomorphic to the known solution. We further analyze the pattern and obtain necessary conditions to further trim down the search space. We conclude with our attempts on finding a 5-coloring of a 26 by 26 grid, as well as further open problems on the shift pattern.

Past work on evacuation planning assumes that evacuees will follow instructions -- however, there is ample evidence that this is not the case. While some people will follow instructions, others will follow their own desires. In this paper, we present a formal definition of a behavior-based evacuation problem (BBEP) in which a human behavior model is taken into account when planning an evacuation. We show that a specific form of constraints can be used to express such behaviors. We show that BBEPs can be solved exactly via an integer program called BB_IP, and inexactly by a much faster algorithm that we call BB_Evac. We conducted a detailed experimental evaluation of both algorithms applied to buildings (though in principle the algorithms can be applied to any graphs) and show that the latter is an order of magnitude faster than BB_IP while producing results that are almost as good on one real-world building graph and as well as on several synthetically generated graphs.

We study the Balanced Connected Subgraph(shortly, BCS) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. Given a vertex-colored graph G=(V,E) , where each vertex in V is colored with either ``red'' or ``blue'', the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is color-balanced, i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.