Computer Science

We resolve the longstanding open problem concerning the computational complexity of Max Cut on interval graphs by showing that it is NP-complete.

Reciprocal best match graphs (RBMGs) are vertex colored graphs whose vertices represent genes and the colors the species where the genes reside. Edges identify pairs of genes that are most closely related with respect to an underlying evolutionary tree. In practical applications this tree is unknown and the edges of the RBMGs are inferred by quantifying sequence similarity. Due to noise in the data, these empirically determined graphs in general violate the condition of being a ``biologically feasible'' RBMG. Therefore, it is of practical interest in computational biology to correct the initial estimate. Here we consider deletion (remove at most k edges) and editing (add or delete at most k edges) problems. We show that the decision version of the deletion and editing problem to obtain RBMGs from vertex colored graphs is NP-hard. Using known results for the so-called bicluster editing, we show that the RBMG editing problem for 2 -colored graphs is fixed-parameter tractable. A restricted class of RBMGs appears in the context of orthology detection. These are cographs with a specific type of vertex coloring known as hierarchical coloring. We show that the decision problem of modifying a vertex-colored graph (either by edge-deletion or editing) into an RBMG with cograph structure or, equivalently, to an hierarchically colored cograph is NP-complete.

We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.

In this work, we analyze a sequential game played in a graph called the Multilevel Critical Node problem (MCN). A defender and an attacker are the players of this game. The defender starts by preventively interdicting vertices (vaccination) from being attacked. Then, the attacker infects a subset of non-vaccinated vertices and, finally, the defender reacts with a protection strategy. We provide the first computational complexity results associated with MCN and its subgames. Moreover, by considering unitary, weighted, undirected, and directed graphs, we clarify how the theoretical tractability of those problems vary. Our findings contribute with new NP-complete, Σ p 2 -complete and Σ p 3 -complete problems. Furthermore, for the last level of the game, the protection stage, we build polynomial time algorithms for certain graph classes.

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) . For a fixed graph H , in the list homomorphism problem, denoted by LHom( H ), we are given a graph G , whose every vertex v is equipped with a list L(v)⊆V(H) . We ask if there exists a homomorphism f from G to H , in which f(v)∈L(v) for every v∈V(G) . Feder, Hell, and Huang [JGT~2003] proved that LHom( H ) is polynomial time-solvable if H is a bi-arc-graph, and NP-complete otherwise. We are interested in the complexity of the LHom( H ) problem in graphs excluding a copy of some fixed graph F as an induced subgraph. It is known that if F is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom( H ) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs F . If F is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if H is not predacious, then for every fixed t the LHom( H ) problem can be solved in quasi-polynomial time in P t -free graphs. On the other hand, if H is predacious, then there exists t , such that LHom( H ) cannot be solved in subexponential time in P t -free graphs. If F is a subdivided claw, we show a full dichotomy in two important cases: for H being irreflexive (i.e., with no loops), and for H being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive H the LHom( H ) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if H is non-predacious and triangle-free. If H is reflexive, then LHom( H ) cannot be solved in subexponential time whenever H is not a bi-arc graph.

The bribery problem in election has received considerable attention in the literature, upon which various algorithmic and complexity results have been obtained. It is thus natural to ask whether we can protect an election from potential bribery. We assume that the protector can protect a voter with some cost (e.g., by isolating the voter from potential bribers). A protected voter cannot be bribed. Under this setting, we consider the following bi-level decision problem: Is it possible for the protector to protect a proper subset of voters such that no briber with a fixed budget on bribery can alter the election result? The goal of this paper is to give a full picture on the complexity of protection problems. We give an extensive study on the protection problem and provide algorithmic and complexity results. Comparing our results with that on the bribery problems, we observe that the protection problem is in general significantly harder. Indeed, it becomes ∑ 2 p -complete even for very restricted special cases, while most bribery problems lie in NP. However, it is not necessarily the case that the protection problem is always harder. Some of the protection problems can still be solved in polynomial time, while some of them remain as hard as the bribery problem under the same setting.

The additively separable hedonic game (ASHG) is a model of coalition formation games on graphs. In this paper, we intensively and extensively investigate the computational complexity of finding several desirable solutions, such as a Nash stable solution, a maximum utilitarian solution, and a maximum egalitarian solution in ASHGs on sparse graphs including bounded-degree graphs, bounded-treewidth graphs, and near-planar graphs. For example, we show that finding a maximum egalitarian solution is weakly NP-hard even on graphs of treewidth 2, whereas it can be solvable in polynomial time on trees. Moreover, we give a pseudo fixed parameter algorithm when parameterized by treewidth.

We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove an existence of a game in this class such that solving the game is PSPACE-hard. The results are based on the construction introduced by Larsson and Wästlund. It relates subtraction games and cellular automata.

This work examines the expected computational cost to determine an approximate global minimum of a class of cost functions characterized by the variance of coefficients. The cost function takes N -dimensional binary states as arguments and has many local minima. Iterations in the order of 2 N are required to determine an approximate global minimum using random search. This work analytically and numerically demonstrates that the selection and crossover algorithm with random initialization can reduce the required computational cost (i.e., number of iterations) for identifying an approximate global minimum to the order of λ N with λ less than 2. The two best solutions, referred to as parents, are selected from a pool of randomly sampled states. Offspring generated by crossovers of the parents' states are distributed with a mean cost lower than that of the original distribution that generated the parents. It is revealed that in contrast to the mean, the variance of the cost of the offspring is asymptotically the same as that of the original distribution. Consequently, sampling from the offspring's distribution leads to a higher chance of determining an approximate global minimum than sampling from the original distribution, thereby accelerating the global search. This feature is distinct from the distribution obtained by a mixture of a large population of favorable states, which leads to a lower variance of offspring. These findings demonstrate the advantage of the crossover between two favorable states over a mixture of many favorable states for an efficient determination of an approximate global minimum.