We present a polynomial-space algorithm that computes the number independent sets of any input graph in time $O(1.1387^n)$ for graphs with maximum degree 3 and in time $O(1.2355^n)$ for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time $O(2.2355^n)$. As a byproduct, we also obtain an exponential-space $O(1.2330^n)$ time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström's compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.
Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice. We introduce $p$-linear colorings as an alternative to the commonly used $p$-centered colorings. They can be efficiently computed in bounded expansion classes and use at most as many colors as $p$-c…
We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an impr…
We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a $(Δ+1)$-coloring of an $n$-node graph with maximum degree $Δ$ in $2^{O(\sqrt{\logΔ})}\cdot\log n$ rounds. This improves on the best previously known time complexity for a large range of values of $Δ$. For graphs with arboricity $a$, we obtain a deterministic distributed algorithm to compute a $(2+o(1))a$-coloring in time $2^{O(\sqrt{\log a})}\cdot\log^2 n$. Further, for graphs with bounded neighborhood independence, we show that a $(Δ+1)$-colorin…
We introduce a variant of the graph coloring problem, which we denote as {\sc Budgeted Coloring Problem} (\bcp). Given a graph $G$, an integer $c$ and an ordered list of integers $\{b_1, b_2, \ldots, b_c\}$, \bcp asks whether there exists a proper coloring of $G$ where the $i$-th color is used to color at most $b_i$ many vertices. This problem generalizes two well-studied graph coloring problems, {\sc Bounded Coloring Problem} (\bocp) and {\sc Equitable Coloring Problem} (\ecp) and as in the case of other coloring problems, it is \nph even for constant values of $c$. So we study \bcp under the…
In this paper we evaluate distributed node coloring algorithms for wireless networks using the network simulator Sinalgo [by DCG@ETHZ]. All considered algorithms operate in the realistic signal-to-interference-and-noise-ratio (SINR) model of interference. We evaluate two recent coloring algorithms, Rand4DColor and ColorReduction (in the following ColorRed), proposed by Fuchs and Prutkin in [SIROCCO'15], the MW-Coloring algorithm introduced by Moscibroda and Wattenhofer [DC'08] and transferred to the SINR model by Derbel and Talbi [ICDCS'10], and a variant of the coloring algorithm of Yu et al.…
We study the 3-\textsc{Coloring} problem in graphs with small diameter. In 2013, Mertzios and Spirakis showed that for $n$-vertex diameter-2 graphs this problem can be solved in subexponential time $2^{\mathcal{O}(\sqrt{n \log n})}$. Whether the problem can be solved in polynomial time remains a well-known open question in the area of algorithmic graphs theory. In this paper we present an algorithm that solves 3-\textsc{Coloring} in $n$-vertex diameter-2 graphs in time $2^{\mathcal{O}(n^{1/3} \log^{2} n)}$. This is the first improvement upon the algorithm of Mertzios and Spirakis in the genera…
The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.
Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W with |B|=b and |W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to permutation graphs.
In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighborhood of a vertex includes itself, and open neighborhoods, where a vertex does not included in its neighborhood. We study the parameterized complexity of conflict-free closed neighborhood coloring and con…
Maximum Partial List H-Coloring on P_5-free graphs in polynomial time
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Warsaw University of Technology（华沙理工大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Institute of Mathematical Sciences, Chennai（金奈数学科学研究所）；University of Bergen（卑尔根大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究院离散数学组）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）
The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrep…
A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors $\mathcal{L}(v)$, for each vertex $v\in V$, $\mathcal{H}$ is said to be $\mathcal{L}$-(list) colorable, if each vertex can be assigned a color from its list such that no edge in $\mathcal{H}$ is monochromatic. We show that list-colorability for any nearly intersecting hypergraph, and lists drawn from a set of constant size, can be checked in quasi-polynomial time in …
The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a kernel of bitsize $O(k^{q-1}\log{k})$ for q-Coloring parameterized by Vertex Cover, for any q >= 3. This size bound is optimal up to $k^{o(1)}$ factors assuming NP is not a subset of coNP/poly, and improve…
For a graph $H$ and integer $k \geq 1$, two functions $f, g$ from $V(H)$ into $\{1, \dots, k\}$ are adjacent if for all edges $uv$ of $H$, $f(u) \neq g(v)$. The graph of all such functions is the exponential graph $K_k^H$. El-Zahar and Sauer proved that if $χ(H) \geq 4$, then $K_3^H$ is 3-chromatic. Tardif showed that, implicit in their proof, is an algorithm for 3-coloring $K_3^H$ whose time complexity is polynomial in the size of $K_3^H$. Tardif then asked if there is an "explicit" algorithm for finding such a coloring: Essentially, given a function $f$ belonging to a 3-chromatic component o…
In an undirected graph, a proper (k,i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k,i)-coloring problem is to compute the minimum number of colors required for a proper (k,i)-coloring. This is a generalization of the classic graph coloring problem. We show a parameterized algorithm for the (k,i)-coloring problem with the size of the feedback vertex set as a parameter. Our algorithm does not use tree-width machinery, thus answering a question of Majumdar, Neogi, Raman and Tale [CALDAM 2017]. We also give a…
This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input gra…
We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm f…
The fastest known classical algorithm deciding the $k$-colorability of $n$-vertex graph requires running time $Ω(2^n)$ for $k\ge 5$. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $O(1.9140^n)$ using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph $20$-coloring problem with running time $O(1.9575^n)$. In the polynomial-space quan…
We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $Δ$ if $k\ge 28$ and $q >357Δ^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931Δ^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\llΔ$ (for large $Δ$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $Δ$ in Moitra's approach.
For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the $H$-Coloring problem the graph $H$ is fixed and we ask whether an instance graph $G$ admits an $H$-coloring. A generalization of this problem is $H$-ColoringExt, where some vertices of $G$ are already mapped to vertices of $H$ and we ask if this partial mapping can be extended to an $H$-coloring. We study the complexity of variants of $H$-Coloring in $F$-free graphs, i.e., graphs excluding a fixed graph $F$ as an induced subgraph. For integers $a,b,c \geq 1$, by $S_{a,b,c}$ we denote the …
An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of the equitable coloring number and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show th…
Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W with |B|=b and |W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to cographs, distance-hereditary graphs, interval graphs and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs.
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $α^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $α>α^*$. We prove that, for any triangle-free graph $G=(V,E)$ with maximum degree $Δ$, for all $q\geqαΔ+1$, the mixing time of the Glaub…
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal sol…
Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring $c:V(G) \rightarrow \{1,2,\ldots,q\}$ such that for each vertex $v \in V(G)$ there is a vertex in $N(v)$ that is uniquely colored from the rest of the vertices in $N(v)$. When we replace $N(v)$ by the closed neighborhood $N[v]$, then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two …
For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists. We also show that this result is tight in some sense, by providing hardness results for the cases where any one of the assumptions does not hold. The edge-coloring variant is also studied, as well as special cases of cographs and split graphs.
In parallel computing, a valid graph coloring yields a lock-free processing of the colored tasks, data points, etc., without expensive synchronization mechanisms. However, coloring is not free and the overhead can be significant. In particular, for the bipartite-graph partial coloring (BGPC) and distance-2 graph coloring (D2GC) problems, which have various use-cases within the scientific computing and numerical optimization domains, the coloring overhead can be in the order of minutes with a single thread for many real-life graphs. In this work, we propose parallel algorithms for bipartite-gra…
We consider the k-strong conflict-free coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflict-free, in the sense that in every interval I there are at least k colors each appearing exactly once in I. In this paper, we present a polynomial algorithm for the general problem; the algorithm has an approximation factor 5-2/k when k\geq2 and approximation factor 2 for k=1. In the special case the family contains all the possible intervals on the given set of points, we show that a 2 approxim…
We study the computational complexity of two related problems: recovering a planted $q$-coloring in $G(n,1/2)$, and finding efficiently verifiable witnesses of non-$q$-colorability (a.k.a. refutations) in $G(n,1/2)$. Our main results show hardness for both these problems in a restricted-but-powerful class of algorithms based on computing low-degree polynomials in the inputs. The problem of recovering a planted $q$-coloring is equivalent to recovering $q$ disjoint planted cliques that cover all the vertices -- a potentially easier variant of the well-studied planted clique problem. Our first re…
We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to $\tO(n^{4/11})$ colors. This is the first combinatorial improvement of Blum's $\tO(n^{3/8})$ bound from FOCS'90. Like Blum's algorithm, our new algorithm composes nicely with recent semi-definite approaches. The current best bound is $O(n^{0.2072})$ colors by Chlamtac from FOCS'07. We now bring it down to $O(n^{0.2038})$ colors.
We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovasz Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations…
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.
