Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

Fine-grained Search Space Classification for Hard Enumeration Variants of Subset Problems

We propose a simple, powerful, and flexible machine learning framework for (i) reducing the search space of computationally difficult enumeration variants of subset problems and (ii) augmenting existing state-of-the-art solvers with informative cues arising from the input distribution. We instantiate our framework for the problem of listing all maximum cliques in a graph, a central problem in network analysis, data mining, and computational biology. We demonstrate the practicality of our approach on real-world networks with millions of vertices and edges by not only retaining all optimal solutions, but also aggressively pruning the input instance size resulting in several fold speedups of state-of-the-art algorithms. Finally, we explore the limits of scalability and robustness of our proposed framework, suggesting that supervised learning is viable for tackling NP-hard problems in practice.Comment: AAAI 201

On the fine-grained complexity of rainbow coloring

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in $k$ colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any $k\ge 2$, there is no algorithm for Rainbow k-Coloring running in time $2^{o(n^{3/2})}$, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set $S$ of pairs of vertices and we ask if there is a coloring in which all the pairs in $S$ are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by $|S|$. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer $q$ and we ask if there is a coloring in which at least $q$ anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by $q$ and has a kernel of size $O(q)$ for every $k\ge 2$ (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]

Suggestions to Improve Lean Construction Planning

The Last Planner System® has been one of the most popular lean construction tools that offers a solution to tackle the problems of production management on construction sites. Since its inception almost 20 years ago, construction companies across the world have implemented Last Planner with reported success. However, even as Last Planner was originally designed to address some shortcomings of the CPM method, a particular shortcoming – namely task continuity was not addressed directly. Also, excepting PPC and Reasons for Non Completion charts, there are no explicit visual tools offered by the Last Planner system. On the other hand, Line of Balance based approaches intrinsically support the consideration of task continuity, and offer a basic visual management approach in schedule representation. With some exceptions, Line of Balance is seen as a special technique applicable only in linear or repetitive work based schedules. The authors suggest that i) there is a need for a robust theory of planning and scheduling and ii) there is a need for a more suitable approach that addresses critical aspects of planning and scheduling function for example by integrating Line of Balance and Last Planner to provide a more robust support for construction scheduling

Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds

We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the number of colors k, the existence of a rainbow s-t path can be decided in O∗ (2k ) time and polynomial space. For the highly related problem of finding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for finding a colorful path in (2 − ε)k nO(1) time for some ε > 0 disproves the so-called Set Cover Conjecture.Second, we focus on the problem of finding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs.Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work

Afroamerikkalaisen urheilijan kapina : ”Olympic Project for Human Rights” ja afroamerikkalaisten näkemykset Philadelphia Tribunen ja Los Angeles Sentinelin kirjoituksissa syyskuusta 1967 elokuuhun 1968

Tutkielma käsittelee kuinka Los Angeles Sentinel ja Philadelphia Tribune kuvailevat afroamerikkalaisten keskuudessa syntyneitä näkemyksiä OPHR:ään liittyen syyskuusta 1967 elokuuhun 1968. OPHR eli ”Olympic Project for Human Rights” oli kansalaisoikeusaktivisti ja sosiologi Harry Edwardsin lokakuussa 1967 perustama organisaatio, joka vastusti rasismia ja pyrki järjestämään afroamerikkalaisten urheilijoiden olympiaboikotin. Boikotti ei lopulta toteutunut, mutta organisaatio pyrkimyksineen sai paljon huomiota. Sentinel ja Tribune ovat molemmat afroamerikkalaisia sanomalehtiä, joiden kirjoittajat olivat myös afroamerikkalaisia. Tutkielmassa tarkastellaan siis ilmiötä afroamerikkalaisesta näkökulmasta. Lähilukua hyödyntämällä ja vertailemalla molempien sanomalehtien toimittajien ja kolumnistien näkemyksiä tuli esille, että OPHR ja olympiaboikotin uhka herättivät vastakkaisia näkemyksiä afroamerikkalaisten keskuudessa kolmen eri teeman kohdalla. Näistä ensimmäinen liittyi urheilun rooliin. Tuohon aikaan eli vielä voimakkaana mielikuva urheilusta suurena tasa-arvoistajana ja moni afroamerikkalainenkin uskoi urheilun edistäneen heidän asemaansa. Tätä käytettiin Sentinelissä ja Tribunessa argumenttina olympiaboikottia vastaan. Toisaalta olympiaboikotin kannattajat huomauttivat, että urheilussa afroamerikkalaiset kohtasivat yhtä lailla rotusyrjintää. Toiseksi afroamerikkalaisten keskuudessa oltiin eri mieltä parhaista keinoista heidän asioidensa ajamiseksi. Osa toimittajista ja kolumnisteista kritisoi OPRH:n toimintaa kuten väkivallalla uhkaamista ja piti olympiaboikottia hyödyttömänä. Toiset taas kehuivat OPHR:ää ja erityisesti nuorten urheilijoiden aktivismia. Vastakkainasettelua taustoittaa se, että osa afroamerikkalaisista ei katsonut kansalaisoikeusliikkeen rauhanomaisen protestoinnin riittävän vaan vaati radikaalimpaa toimintaa. Mustan vallan liike vaikutti merkittävästi Edwardsin ajattelun ja OPHR:n toiminnan taustalla. Lisäksi Sentinelissä ja Tribunessa kirjoitettiin Edwardsista eri tavoin. Edwards sai kiitosta afroamerikkalaisten kohtaaman syrjinnän esille nostamisesta. Edwards oli kuitenkin myös provosoiva puheissaan ja olemukseltaan ja hän kohtasi myös paljon kritiikkiä