Ising machines can find optimum or quasi-optimum solutions of combinatorial optimization problems efficiently and effectively. The graph coloring problem, which is one of the difficult combinatorial optimization problems, is to assign a color to each vertex of a graph such that no two vertices connected by an edge have the same color. Although methods to map the graph coloring problem onto the Ising model or quadratic unconstrained binary optimization (QUBO) model are proposed, none of them considers minimizing the number of colors. In addition, there is no Ising-machine-based method considering additional constraints in order to apply to practical problems. In this paper, we propose a mapping method of the graph coloring problem including minimizing the number of colors and additional constraints to the QUBO model. As well as the constraint terms for the graph coloring problem, we firstly propose an objective function term that can minimize the number of colors so that the number of used spins cannot increase exponentially. Secondly, we propose two additional constraint terms: One is that specific vertices have to be colored with specified colors; The other is that specific colors cannot be used more than the number of times given in advance. We theoretically prove that, if the energy of the proposed QUBO mapping is minimized, all the constraints are satisfied and the objective function is minimized. The result of the experiment using an Ising machine showed that the proposed method reduces the number of used colors by up to 75.1% on average compared to the existing baseline method when additional constraints are not considered. Considering the additional constraints, the proposed method can effectively find feasible solutions satisfying all the constraints.

Inter-channel correlation is one of the redundancy which need to be eliminated in video coding. In the latest video coding standard H.266/VVC, the DM (Direct Mode) and CCLM (Cross-component Linear Model) modes have been introduced to reduce the similarity between luminance and chroma. However, inter-channel correlation is still observed. In this paper, a new inter-channel prediction algorithm is proposed, which utilizes coloring principle to predict chroma pixels. From the coloring perspective, for most natural content video frames, the three components Y, U and V always demonstrate similar coloring pattern. Therefore, the U and V components can be predicted using the coloring pattern of the Y component. In the proposed algorithm, correlation coefficients are obtained in a lightweight way to describe the coloring relationship between current pixel and reference pixel in Y component, and used to predict chroma pixels. The optimal position for the reference samples is also designed. Base on the selected position of the reference samples, two new chroma prediction modes are defined. Experiment results show that, compared with VTM 12.1, the proposed algorithm has an average of -0.92% and -0.96% BD-rate improvement for U and V components, for All Intra (AI) configurations. At the same time, the increased encoding time and decoding time can be ignored.

The Graph Coloring Problem (GCP) is a fundamental combinatorial optimization problem that has many practical applications. Degree of SATURation (DSATUR) and Recursive Largest First (RLF) are well known as typical solution construction algorithms for GCP. It is necessary to update the vertex degree in the subgraph induced by uncolored vertices when selecting vertices to be colored in both DSATUR and RLF. There is an issue that the higher the edge density of a given graph, the longer the processing time. The purposes of this paper are to propose a degree updating method called Adaptive Degree Updating (ADU for short) that improves the issue, and to evaluate the effectiveness of ADU for DSATUR and RLF on DIMACS benchmark graphs as well as random graphs having a wide range of sizes and densities. Experimental results show that the construction algorithms with ADU are faster than the conventional algorithms for many graphs and that the ADU method yields significant speed-ups relative to the conventional algorithms, especially in the case of large graphs with higher edge density.

We study the problem of transforming one (vertex) c-coloring of a graph into another one by changing only one vertex color assignment at a time, while at all times maintaining a c-coloring, where c denotes the number of colors. This decision problem is known to be PSPACE-complete even for bipartite graphs and any fixed constant c ≥ 4. In this paper, we study the problem from the viewpoint of graph classes. We first show that the problem remains PSPACE-complete for chordal graphs even if c is a fixed constant. We then demonstrate that, even when c is a part of input, the problem is solvable in polynomial time for several graph classes, such as k-trees with any integer k ≥ 1, split graphs, and trivially perfect graphs.

Let G be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. Suppose that we are given two list edge-colorings f0 and fr of G, and asked whether there exists a sequence of list edge-colorings of G between f0 and fr such that each list edge-coloring can be obtained from the previous one by changing a color assignment of exactly one edge. This problem is known to be PSPACE-complete for every integer k ≥ 6 and planar graphs of maximum degree three, but any computational hardness was unknown for the non-list variant in which every edge has the same list of k colors. In this paper, we first improve the known result by proving that, for every integer k ≥ 4, the problem remains PSPACE-complete even for planar graphs of bounded bandwidth and maximum degree three. Since the problem is known to be solvable in polynomial time if k ≤ 3, our result gives a sharp analysis of the complexity status with respect to the number k of colors. We then give the first computational hardness result for the non-list variant: for every integer k ≥ 5, the non-list variant is PSPACE-complete even for planar graphs of bandwidth quadratic in k and maximum degree k.

The online interval coloring problem has been extensively studied for many years. Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for this problem. The number of colors used by the algorithm is at most 3ω(G)-2, where ω(G) is the size of the maximum clique in a given graph G. Also, they presented an instance for which the number of colors used by any online algorithm is at least 3ω(G)-2. This instance includes intervals with various lengths, which cannot be applied to the case when the lengths of the given intervals are restricted to one, i.e., the online unit interval coloring problem. In this case, the current best upper and lower bounds on the number of colors used by an online algorithm are 2ω(G)-1 and 3ω(G)/2 respectively by Epstein and Levy (ICALP2005). In this letter, we conduct a complete performance analysis of the Kierstead-Trotter algorithm for online unit interval coloring, and prove it is NOT optimal. Specifically, we provide an upper bound of 3ω(G)-3 on the number of colors used by their algorithm. Moreover, the bound is the best possible.

Buffer overflow is one of the main approaches to get control of vulnerable programs. This paper presents a protection technique called BFWindow for performance and resource sensitive embedded systems. By coloring data structure in memory with single associate property bit to each byte and extending the target memory block to a BFWindow(2), it validates each memory write by speculatively checking consistency of data properties within the extended buffer window. Property bits are generated by compiler statically and checked by hardware at runtime. They are transparent to users. Experimental results show that the proposed mechanism is effective to prevent sequential memory writes from crossing buffer boundaries which is the common scenario of buffer overflow exploitations. The performance overhead for practical protection mode across embedded system benchmarks is under 1%.

The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic number is NP-hard even for trees with pathwidth at most 2. Hence, we consider the harmonious coloring of trees with pathwidth 1, which are also known as caterpillars. This paper shows the harmonious chromatic number of a caterpillar with at most one vertex of degree more than 2. We also show the upper bound of the harmonious chromatic number of a 3-regular caterpillar.

We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACE-completeness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives sharp analyses of the problem with respect to pathwidth.

In digital cinema, an image goes through many types of processes like scanning, mastering, and digital intermediate. Among them, the digital intermediate process plays a central role because it determines the final color of an image. It edits and changes the colors of the images. However, some color distortions such as color bleeding are generated when editing and changing local colors in an image. In this paper, local color improvement for digital intermediate is proposed based on color transfer. Our method is simple and efficient color improvement that does not requires neither precise image segmentation nor feature matching. To prevent color distortions, a modified color influence map is proposed with color categories. First, the source image is roughly segmented using a color category map, which groups similar colors in color space. Second, the color influence map is modified by assigning different weights to the lightness and chroma components. Lastly, the modified color influence map and color category map filtered with anisotropic diffusion are combined. Experimental results show that the proposed method produces less color distortion in the resulting image.

An orthogonal ray graph is an intersection graph of horizontal and vertical rays (closed half-lines) in the plane. Such a graph is 3-directional if every vertical ray has the same direction, and 2-directional if every vertical ray has the same direction and every horizontal ray has the same direction. We derive some structural properties of orthogonal ray graphs, and based on these properties, we introduce polynomial-time algorithms that solve the dominating set problem, the induced matching problem, and the strong edge coloring problem for these graphs. We show that for 2-directional orthogonal ray graphs, the dominating set problem can be solved in O(n2 log5 n) time, the weighted dominating set problem can be solved in O(n4 log n) time, and the number of dominating sets of a fixed size can be computed in O(n6 log n) time, where n is the number of vertices in the graph. We also show that for 2-directional orthogonal ray graphs, the weighted induced matching problem and the strong edge coloring problem can be solved in O(n2+m log n) time, where m is the number of edges in the graph. Moreover, we show that for 3-directional orthogonal ray graphs, the induced matching problem can be solved in O(m2) time, the weighted induced matching problem can be solved in O(m4) time, and the strong edge coloring problem can be solved in O(m3) time. We finally show that the weighted induced matching problem can be solved in O(m6) time for orthogonal ray graphs.

In dense femtocell networks (DFNs), one of the main issues is interference management since interference between femtocell access points (FAPs) reduces the system performance significantly. Further, FAPs serve different numbers of femtocell user equipments (FUEs), i.e., some FAPs have more than one FUE while others have one or no FUEs. Therefore, for DFNs, an intelligent channel assignment scheme is necessary considering both the number of FUEs connected to the same FAPs and interference mitigation to improve system performance. This paper proposes a two-stage dynamic channel assignment (TS-DCA) scheme for downlink DFNs based on orthogonal frequency division multiple access/frequency division duplex (OFDMA/FDD). In stage 1, using graph coloring algorithm, a femtocell gateway (FGW) first groups FUEs based on an interference graph that considers different numbers of FUEs per FAP. Then, in stage 2, the FGW dynamically assigns subchannels to FUE clusters according to the order of maximum capacity of FAP clusters. In addition, FAPs adaptively assign remaining subchannels in FUE clusters to their FUEs in other FUE clusters. Through simulations, we first find optimum parameters of the FUE clustering to maximize the system capacity and then evaluate system performance in terms of the mean FUE capacity, unsatisfied FUE probability, and mean FAP transmission energy consumption according to the different numbers of FUEs and FAPs with a given FUE traffic load.

Many problems can be formulated as maximum clique problems. Hence, it is highly important to develop algorithms that can find a maximum clique very fast in practice. We propose new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, pp.95–111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Extensive computational experiments confirm the superiority of MCS over MCR and other existing algorithms. It is faster than the other algorithms by orders of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graph. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs. This paper demonstrates in detail the effectiveness of each new techniques in MCS, as well as the overall contribution.

A method obtaining a monochrome image which can rebuild colors is proposed. In this method, colors in an input image are quantized under a lightness constraint and a palette, which represents relationship between quantized colors and gray-levels, is generated. Using the palette, an output monochrome image is obtained. Experiments show that the proposed method obtains good monochrome and rebuilt color images.

We study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing only one edge color assignment at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. Ito, Kamiski and Demaine gave a sufficient condition so that any list edge-coloring of a tree can be transformed into any other. In this paper, we give a new sufficient condition which improves the known one. Our sufficient condition is best possible in some sense. The proof is constructive, and yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices via O(n2) recoloring steps. We remark that the upper bound O(n2) on the number of recoloring steps is tight, because there is an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n2) recoloring steps.

Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(f) of an edge-coloring f of G is the sum of costs ω(f(e)) of colors f(e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω(f) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog(nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.

The planar Hajos calculus (PHC) is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajos calculus (PHC(dd)) is PHC with the restriction that all the graphs that appear in the construction (including a final graph) must have maximum degree at most d. We prove the followings: (1) If PHC is polynomially bounded, then for any d ≥ 4, PHC(dd+2) can generate any non-3-colorable planar graphs of maximum degree at most d in polynomial steps. (2) If PHC can generate any non-3-colorable planar graphs of maximum degree 4 in polynomial steps, then PHC is polynomially bounded.

In this letter we show that the dynamic optimal PCID allocation problem in LTE systems is NP-complete. Subsequently we provide a near-optimal solution using SON which models the problem using new merge operations and explores the search space using a suitable randomized algorithmic approach. Two feasible options for dynamic auto-configuration of the system are also discussed. Simulation results point out that the approach provides near-optimal auto-configuration of PCIDs in computationally feasible time.

We propose an interactive method for assisting the coloring process of 2D hand-drawn animated cartoons. It segments input frames (each hand-drawn drawing of the cartoon) into regions (areas surrounded by closed lines. E.g. the head, the hands) extracts their features, and then matches the regions between frames, allowing the user to fix coloring mistakes interactively. Its main contribution consists in storing matched regions in lists called "chains" for tracking how the region features vary along the animation. Consequently, the matching rate is improved and the matching mistakes are reduced, thus reducing the total effort needed until having a correctly colored cartoon.

The Hajos calculus is a nondeterministic procedure which generates the class of non-3-colorable graphs. If all non-3-colorable graphs can be constructed in polynomial steps by the calculus, then NP = co-NP holds. Up to date, however, it remains open whether there exists a family of graphs that cannot be generated in polynomial steps. To attack this problem, we propose two graph calculi PHC and PHC* that generate non-3-colorable planar graphs, where intermediate graphs in the calculi are also restricted to be planar. Then we prove that PHC and PHC* are sound and complete. We also show that PHC* can polynomially simulate PHC.