In the covering version of the pinwheel scheduling problem, a daily task must be assigned to agents under the constraint that agent i can perform the task at most once in any a_i-day interval. In this paper, we determine the optimal constant α^* = 1.264… such that every instance with ∑_i 1/a_i ≥ α^* is schedulable. This resolves an open problem posed by Kawamura and Soejima (2020). Our proof combines Kawamura’s (2026) techniques for the packing version with new mathematical insights to reduce the analysis to a finite set of instances, which are then verified through an exhaustive computer-aided search that draws on ideas from Gąsieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).

In the 2-Vertex-Connected Spanning Subgraph problem (2-VCSS), we are given an undirected graph G, and the objective is to find a 2-vertex-connected spanning subgraph S of G with the minimum number of edges. In the context of survivable network design, 2-VCSS is one of the most fundamental and well-studied problems. There has been active research on improving the approximation ratio of algorithms, and the current best ratio is 4/3, achieved by Bosch-Calvo, Grandoni, and Jabal Ameli. In this paper, we improve the approximation ratio to 95/72+ε (< 1.32). The key idea in our algorithm is to introduce a 2-edge-cover without certain cycle components, and use it as an initial solution.

Abstract. We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. Whether the combinatorial (i.e., graph-theoretic) distance between vertices of the associahedron can be computed in polynomial time is a tantalizing and important open problem, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that an approach for this open problem is not promising if it is applicable to the generalized problem on graph associahedra. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless [Formula: see text]. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids.

Abstract. The problem of finding a maximum 2-matching without short cycles has received significant attention due to its relevance to the Hamiltonian cycle problem. This problem is generalized to finding a maximum [Formula: see text]-matching which excludes specified complete [Formula: see text]-partite subgraphs, where [Formula: see text] is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete [Formula: see text]-partite subgraphs are edge-disjoint, and in the second case the maximum degree of the input graph is at most [Formula: see text]. Our result for the first case extends the previous work of Nam (1994), showing the polynomial solvability of the problem of finding a maximum 2-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of Bérczi and Végh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most [Formula: see text]. Our algorithms are obtained from exploiting the discrete structure of restricted [Formula: see text]-matchings and employing an algorithm for the Boolean edge-CSP.

We study the electrical distribution network reconfiguration problem, defined as follows. We are given an undirected graph with a root vertex, demand at each non-root vertex, and resistance on each edge. Then, we want to find a spanning tree of the graph that specifies the routing of power from the root to each vertex so that all the demands are satisfied and the energy loss is minimized. This problem is known to be NP-hard in general. When restricted to grids with uniform resistance and the root located at a corner, Gupta, Khodabaksh, Mortagy and Nikolova [Mathematical Programming 2022] invented the so-called Min-Min algorithm whose approximation factor is theoretically guaranteed. Our contributions are twofold. First, we prove that the problem is NP-hard even for grids; this resolves the open problem posed by Gupta et al. Second, we give a refined analysis of the Min-Min algorithm and improve its approximation factor under the same setup. In the analysis, we formulate the problem of giving an upper bound for the approximation factor as a non-linear optimization problem that maximizes a convex function over a polytope.

The minimum sum coloring problem with bundles was introduced by Darbouy and Friggstad (SWAT 2024) as a common generalization of the minimum coloring problem and the minimum sum coloring problem. During their presentation, the following open problem was raised: whether the minimum sum coloring problem with bundles could be solved in polynomial time for trees. We answer their question in the negative by proving that the minimum sum coloring problem with bundles is NP-hard even for paths. We complement this hardness by providing algorithms of the following types. First, we provide a fixed-parameter algorithm for trees when the number of bundles is a parameter; this can be extended to graphs of bounded treewidth. Second, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and the difference between the number of vertices and the number of bundles is constant. Third, we provide a polynomial-time algorithm for trees when bundles form a partition of the vertex set and each bundle induces a connected subgraph. We further show that for bipartite graphs, the problem with weights is NP-hard even when the number of bundles is at least three, but is polynomial-time solvable when the number of bundles is at most two. The threshold shifts to three versus four for the problem without weights.

Bérczi K., Király T., Kobayashi Y., et al. Finding spanning trees with perfect matchings. Discrete Applied Mathematics 371, 137 (2025); https://doi.org/10.1016/j.dam.2025.04.001.

Abstract. We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate concepts of proportionality have been considered in previous work. Among them, proportionality up to the maximin good ( PROPm ) has been the best approximate notion of proportionality that can be achieved for all instances [A. Baklanov et al., PROPm allocations of indivisible goods to multiple agents, in Proceedings of the 30th International Joint Conference on Artificial Intelligence, 2021, pp. 24–30]. In this paper, we introduce the notion of proportionality up to the least valued good on average ( PROPavg ), which is a stronger notion than PROPm , and show that a PROPavg allocation always exists for all instances and can be computed in polynomial time. Our results establish PROPavg as a notable nontrivial fairness notion that can be achieved for all instances. Our proof is constructive and is based on a new technique that generalizes the cut-and-choose protocol and uses a recursive technique.

In this article, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths \(\mathcal{P}\) and \(\mathcal{Q}\) connecting the same set of terminal pairs, we aim to determine whether \(\mathcal{P}\) can be transformed to \(\mathcal{Q}\) by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration . We first show that Disjoint Paths Reconfiguration is \(\mathsf{PSPACE}\) -complete even when \(k=2\) . On the other hand, we prove that, when the graph is embedded on a plane and all paths in \(\mathcal{P}\) and \(\mathcal{Q}\) connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s - t paths as a variant. We show that the disjoint s - t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is \(\mathsf{PSPACE}\) -complete in general.

The qubit routing problem, also known as the swap minimization problem, is a (classical) combinatorial optimization problem that arises in the design of compilers of quantum programs. We study the qubit routing problem from the viewpoint of theoretical computer science, while most of the existing studies investigated the practical aspects. We concentrate on the linear nearest neighbor (LNN) architectures of quantum computers, in which the graph topology is a path. Our results are three-fold. (1) We prove that the qubit routing problem is NP-hard. (2) We show that the qubit routing problem on a path can be solved in a fixed-parameter time where the number of two-qubit gates is a parameter. (3) We show that the qubit routing problem on a path can be solved in polynomial time if each qubit is involved in at most one two-qubit gate.

In fair division problems, we are given a set S of m items and a set N of n agents with individual preferences, and the goal is to find an allocation of items among agents so that each agent finds the allocation fair. There are several established fairness concepts and envy-freeness is one of the most extensively studied ones. However envy-free allocations do not always exist when items are indivisible and this has motivated relaxations of envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are two well-studied relaxations. We consider the problem of finding EF1 and EFX allocations for utility functions that are not necessarily monotone, and propose four possible extensions of different strength to this setting. In particular, we present a polynomial time algorithm for finding an EF1 allocation for two agents with arbitrary utility functions. An example is given showing that EFX allocations need not exist for two agents with non-monotone, non-additive, identical utility functions. However, when all agents have monotone (not necessarily additive) identical utility functions, we give a pseudo-polynomial time algorithm that always finds an EFX allocation of chores. As a step toward understanding the general case, we discuss two subclasses of utility functions: Boolean utilities that are {0,+1}-valued functions, and negative Boolean utilities that are {0,−1}-valued functions. For the latter, we give a polynomial time algorithm that finds an EFX allocation when the utility functions are identical.

Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only ``approximately'' formulated for original real-world problems. To solve this issue, finding \emph{multiple} solutions is a natural direction, and diversity of solutions is an important concept in this context. Unfortunately, finding diverse solutions is much harder than finding a single solution. To cope with the difficulty, we investigate the approximability of finding diverse solutions. As a main result, we propose a framework to design approximation algorithms for finding diverse solutions, which yields several outcomes including constant-factor approximation algorithms for finding diverse matchings in graphs and diverse common bases in two matroids and PTASes for finding diverse minimum cuts and interval schedulings.

In this paper, we consider a transformation of $k$ disjoint paths in a graph. For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$ connecting the same set of terminal pairs, we aim to determine whether $\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one path with another path so that the intermediates are also $k$ disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when $k=2$. On the other hand, we prove that, when the graph is embedded on a plane and all paths in $\mathcal{P}$ and $\mathcal{Q}$ connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint $s$-$t$ paths as a variant. We show that the disjoint $s$-$t$ paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal the reconfigurability of the union of $k$ arborescences for fixed $k$ in the following sense: for any pair of arc subsets that can be partitioned into $k$ arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into $k$ arborescences. This generalizes the result by Ito et al. (2023), who showed the case with $k=1$. Since the union of $k$ arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

The $2$-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. In the problem, we are given an undirected graph $G$, and the objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with factor $1.326$, which was the best approximation ratio. In this paper, we give a $(1.3+\varepsilon)$-approximation algorithm for 2-ECSS, where $\varepsilon$ is an arbitrary positive fixed constant, which improves the previously known best approximation ratio. In our algorithm, we compute a minimum triangle-free $2$-edge-cover in $G$ with the aid of the algorithm for finding a maximum triangle-free $2$-matching given by Hartvigsen. Then, with the obtained triangle-free $2$-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli.

We initiate the study of k -edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k -edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k -edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k -edge-connected orientation if and only if G is 2k -edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k -edge-connected orientations of an undirected graph G is connected if G is (2k+2) -edge connected. This has been known to be true only when k=1 .

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

Allocating resources to individuals in a fair manner has been a topic of interest since the ancient times, with most of the early rigorous mathematical work on the problem focusing on infinitely divisible resources. Recently, there has been a surge of papers studying computational questions regarding various different notions of fairness for the indivisible case, like maximin share fairness (MMS) and envy-freeness up to any good (EFX). We survey the most important results in the discrete fair division literature, focusing on the case of additive valuation functions and paying particular attention to the progress made in the last 10 years.

We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is k-edge-connected. This yields an “edge-flip based” new proof of Nash-Williams' theorem: an undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k + 2)-edge-connected. This has been known to be true only when k = 1.

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.

Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly $$\ell $$ ℓ that have endpoints in A and internal points in $$V{\setminus }A$$ V \ A . We study the parameterized complexity of this problem with parameters |A|, $$\ell $$ ℓ , k, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $$\ell \le 3$$ ℓ ≤ 3 , while it is NP-complete for constant $$\ell \ge 4$$ ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth $${}+|A|$$ + | A | , while it is fixed-parameter tractable parameterized by treewidth $${}+\ell $$ + ℓ . Additionally, we study a variant called Short A-Path Packing that asks to find k vertex-disjoint paths of length at most $$\ell $$ ℓ . We show that all our positive results on the exact-length version can be translated to this version and show the hardness of the cases where |A| or $$\ell $$ ℓ is a constant.

Abstract A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovász [ Acta Sci. Math., 42 (1980), pp. 121--131] showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. In this paper, we present a combinatorial, deterministic, polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polynomial matrix formulation using Pfaffian and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann [ Combinatorica, 6 (1986), pp. 123--150] for the unweighted problem.

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable of achieving optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with matroid rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by Dütting and Végh [Private communication, 2017]. We further formalize a weighted variant of the conjecture of Dütting and Végh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions or, equivalently, for gross substitutes functions.

Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem of determining, given two spanning trees $T_1,T_2$ with an additional property $Π$, if there exists an edge flip transformation from $T_1$ to $T_2$ keeping property $Π$ all along. First we show that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at most $k$ (for any fixed $k \ge 3$) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at least $k$ leaves (where $k$ is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when $k=n-2$.

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by Düetting and Végh in 2017. We further formalize a weighted variant of the conjecture of Düetting and Végh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions.

Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures PSPACE and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar AND/OR graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of AND or OR vertices of an AND/OR constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of AND vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing PSPACE.

For a positive integer $t$ and a graph $G$, an additive $t$-spanner of $G$ is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus $t$. Minimum Additive $t$-Spanner Problem is to find an additive $t$-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive $t$-spanners, Minimum Additive $t$-Spanner Problem is hard to handle, and hence only few results are known for it. In this paper, we study Minimum Additive $t$-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to $(α, β)$-spanners.

A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.

We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in O(3.460^k n) time, where k is the solution size. This improves the previous best O(3.619^k n)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf. Process. Lett., 2014).

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

We study the following all-or-nothing multicommodity flow problem in planar graphs: Input: A graph $G$ with $n$ vertices and $k$ pairs of vertices $(s_1,t_1),(s_2,t_2),\dots, (s_k,t_k)$ in $G$. Find: A largest subset $W$ of $\{1,\dots,k\}$ such that for every $i$ in $W$, we can send one unit of flow between $s_i$ and $t_i$. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri, Khanna, and Shepherd [ SIAM J. Comput., 42 (2013), pp. 1467--1493] give a polylogarithmic factor approximation algorithm and show that a natural LP-relaxation has a polylogarithmic integrality gap. This result is in contrast with the integrality gap $\Omega(\sqrt{n})$ for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an $O(1)$-approximation algorithm and show that the integrality gap is $O(1)$. In particular, in polynomial time, we can find an index set $W$ with $|W| = \Omega({OPT})$ and eight $s_i$-$t_i$ paths for each $i \in W$ such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the result by Séguin-Charbonneau and Shepherd [ Proceedings of FOCS, 2011, pp. 200--209], who give an $O(1)$-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

In the k disjoint shortest paths problem (k-DSPP), we are given a graph and its vertex pairs (s_1, t_1), ... , (s_k, t_k), and the objective is to find k pairwise disjoint paths P_1, ... , P_k such that each path P_i is a shortest path from s_i to t_i, if they exist. If the length of each edge is equal to zero, then this problem amounts to the disjoint paths problem, which is one of the well-studied problems in algorithmic graph theory and combinatorial optimization. Eilam-Tzoreff (1998) focused on the case when the length of each edge is positive, and showed that the undirected version of 2-DSPP can be solved in polynomial time. Polynomial solvability of the directed version was posed as an open problem by Eilam-Tzoreff (1998). In this paper, we solve this problem affirmatively, that is, we give a first polynomial time algorithm for the directed version of 2-DSPP when the length of each edge is positive. Note that the 2 disjoint paths problem in digraphs is NP-hard, which implies that the directed 2-DSPP is NP-hard if the length of each edge can be zero. We extend our result to the case when the instance has two terminal pairs and the number of paths is a fixed constant greater than two. We also show that the undirected k-DSPP and the vertex-disjoint version of the directed k-DSPP can be solved in polynomial time if the input graph is planar and k is a fixed constant.

The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars.

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Lovász (1980) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem.

In this paper, we show the Erdös--Pósa property for edge-disjoint packing of $S$-closed walks with parity constraints in 4-edge-connected graphs. More precisely, we prove that for any 4-edge-connected graph $G$ and any vertex subset $S$, either $G$ has $k$ edge-disjoint elementary closed odd walks, each of which has at least one vertex of $S$, or $G$ has an edge set $F$ with $|F| \leq f(k)$ such that $G-F$ has no such walks. The 4-edge-connectivity is the best possible in the sense that 3-edge-connected graphs do not satisfy the statement. Since the proof is constructive, we can design a fixed-parameter algorithm for finding $k$ edge-disjoint walks satisfying the conditions in a 4-edge-connected graph for a parameter $k$. In addition, this gives a simple fixed-parameter algorithm for the parity edge-disjoint walks problem with $k$ terminal pairs.

Edmonds's fundamental theorem on arborescences in [J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithms, Courant Comput. Sci. Sympos. 9, Algorithmics Press, New York, 1973, pp. 91--96] characterizes the existence of $k$ pairwise arc-disjoint spanning arborescences with the same root in a directed graph. In [L. Lovász, J. Combinatorial Theory Ser. B, 21 (1976), pp. 96--103], Lovász gave an elegant alternative proof which became the basis of many extensions of Edmonds's result. In this paper, we use a modification of Lovász's method to prove a theorem on covering intersecting bi-set families under matroid constraints. Our result can be considered as an extension of previous results on packing arborescences. We also investigate the algorithmic aspects of the problem and present a polynomial-time algorithm for solving the corresponding optimization problem.

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r -approximation algorithm for this problem is a polynomial-time algorithm that finds at least OPT/ r edge-disjoint paths, where OPT denotes the maximum possible number of pairs that can be routed in a given instance. For a long time, an O ( n 1/2 -approximation algorithm has been best known for this problem even if a congestion of two is allowed, that is, each edge is allowed to be used in at most two of the paths. In this article, we give a randomized O ( n 3/7 ċ poly(log n ))-approximation algorithm with congestion two. This is the first result that breaks the O ( n 1/2 )-approximation algorithm. In particular, we prove the following: (1) If we have a (randomized) polynomial-time algorithm for finding Ω(OPT1/p /polylog( n )) edge-disjoint paths for some p > 1, then we can give a randomized O ( n 1/2 -α)-approximation algorithm for the edge-disjoint paths problem by using Rao-Zhou’s algorithm for some α > 0. (2) Based on the Chekuri-Khanna-Shepherd well-linked decomposition, we show that there is a randomized algorithm for finding Ω(OPT 1/4 /(log n ) 3/2 ) edge-disjoint paths connecting given terminal pairs with congestion two. Our framework for this algorithm is more general in the following sense. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if there is a (randomized) polynomial-time algorithm for finding Ω(OPT1/p) edge-disjoint paths connecting given terminal pairs for some p > 1.

We study variants of the vertex disjoint paths problem in plane graphs where paths have to be selected from given sets of paths. We investigate the problem as a decision, maximization, and routing-in-rounds problem. Although all considered variants are NP-hard in planar graphs, restrictions on the locations of the terminals on the outer face of the given planar embedding of the graph lead to polynomially solvable cases for the decision and maximization versions of the problem. For the routing-in-rounds problem, we obtain a p-approximation algorithm, where p is the maximum number of alternative paths for a terminal pair, when restricting the locations of the terminals to the outer face such that they appear in a counterclockwise traversal of the boundary as a sequence for some permutation . © 2015 Wiley Periodicals, Inc.NETWORKS, Vol. 66(2), 136–144 2015

We consider the following network design problem that we call the Generalized Terminal Backup Problem: given a graph (or a hypergraph) $G_0=(V, {E}_0)$, a set of (at least 2) terminals $T\subseteq V$, and a requirement $r(t)$ for every $t\in T$, find a multigraph $G=(V,E)$ such that $\lambda_{G_0+G}(t, T-t)\ge r(t)$ for any $t\in T$. In the minimum cost version the objective is to find $G$ minimizing the total cost $c(E)=\sum_{uv\in E}c(uv)$, given also costs $c(uv)\ge 0$ for every pair $u,v\in V$. In the degree-specified version the question is to decide whether such a $G$ exists, satisfying that the degree of $v \in V$ is a prescribed value $m(v)$. The Terminal Backup Problem solved in [E. Anshelevich and A. Karagiozova, SIAM J. Comput., 40 (2011), pp. 678--708] is the special case where $G_0$ is the empty graph and $r(t)=1$ for every terminal $t\in T$. We solve the Generalized Terminal Backup Problem in the following two cases. In the first case we solve the degree-specified version by a splitting-off theorem. This splitting-off theorem in turn provides the solution for the minimum cost version in the case when $c$ is node-induced, that is $c(uv)=w(u)+w(v)$ for some node weights $w:V\to \mathbb{R}_+$. In the second case we turn to the general minimum cost version, and we are able to solve it when $G_0$ is the empty graph. This includes the Terminal Backup Problem ($r\equiv 1$) and the Maximum-Weight $b$-matching Problem ($T=V$). The solution depends on an interesting new variant of a theorem of Lovász and Cherkassky, and on the solution of the so-called Simplex Matching Problem. Our algorithms run in polynomial time for both problems.

Fault-tolerance is one of the most important factors in designing networks. Failures in networks are sometimes caused by an event occurring in specific geographical regions such as hurricanes, earthquakes, bomb attacks, and Electromagnetic Pulse (EMP) attacks. In INFOCOM 2012, Neumayer et al. introduced geographical variants of max-flow min-cut problems in a circular disk failure model, in which each failure is represented by a disk with a predetermined size. In this paper, we solve two open problems in this model: we give a first polynomial-time algorithm for the geographic max-flow problem, and prove a conjecture of Neumayer et al. on a relationship between the geographic max-flow and the geographic min-cut.

We consider the following network design problem, that we call the Generalized Terminal Backup Problem. Given a graph (or a hypergraph) G0 = (V, E0), a set of (at least 2) terminals T ⊆ V and a requirement r(t) for every t ∊ T, find a multigraph G = (V, E) such that for any t ∊ T. In the minimum cost version the objective is to find G minimizing the total cost c(E) = Σuv∊E c(uv), given also costs c(uv) ≥ 0 for every pair u, v ∊ V. In the degree-specified version the question is to decide whether such a G exists, satisfying that the number of edges is a prescribed value g(v) at each node v ∊ V. The Terminal Backup Problem solved in [1] is the special case where G0 is the empty graph and r(t) = 1 for every terminal t ∊.T. We solve the Generalized Terminal Backup Problem in the following two cases. 1. In the first case we start with the minimum cost version for c ≡ 1, which helps solving the degree-specified version by a splitting-off theorem. This splitting-off theorem in turn provides the solution for the minimum cost version in the case when c is node-induced, that is c(uv) = w(u) + w(v) for some node weights w : V → ℝ+. The algorithm for this case is polynomial. 2. In the second solved case we turn to the general minimum cost version, and we are able to solve it when G0 is the empty graph. This includes the Terminal Backup Problem [1] (r ≡ 1) and the Maximum-Weight b-matching Problem (T = V). The solution depends on an interesting new variant of a theorem of Lovász and Cherkassky, and on the solution of the so-called Simplex Matching problem [1]. Our algorithm is polynomial in |V| and max{r(t) : t ∊ T}.

The excluded grid theorem, originally proved by Robertson and Seymour in Graph Minors V, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance as the basis for bidimensionality theory on graph classes excluding a fixed minor.

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between siand ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-tipaths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

In a weighted independence system, an independent set is said to be $\alpha$-robust if, for all $p$, the total weight of its heaviest $p$ elements is at least $\alpha$ times the maximum weight of a $p$-independent set. Here a $p$-independent set is an independent set with at most $p$ elements. The set of matchings in a weighted graph is a typical example of a weighted independence system, and Hassin and Rubinstein [SIAM J. Discrete Math., 15 (2002), pp. 530--537] showed that every graph has a $\frac{1}{\sqrt2}$-robust matching and it can be found by a $k$th power algorithm in polynomial time. In this paper, we show that it can be extended to the matroid intersection problem; i.e., there always exists a $\frac{1}{\sqrt2}$-robust matroid intersection, which is polynomially computable. We also study the time complexity of the robust matching problem. We show that a 1-robust matching can be computed in polynomial time (if one exists), and, for any fixed number $\alpha$ with $\frac{1}{\sqrt2}<\alpha<1$, the problem to determine whether a given weighted graph has an $\alpha$-robust matching is NP-complete. These together with the positive result for $\alpha=\frac{1}{\sqrt2}$ in [R. Hassin and S. Rubinstein, SIAM J. Discrete Math., 15 (2002), pp. 530--537] give us a sharp border for the complexity for the robust matching problem. Moreover, we show that the problem is strongly NP-complete when $\alpha$ is a part of the input. Finally, we show the limitations of the $k$th power algorithm for robust matchings; i.e., for any $\epsilon>0$, there exists a weighted graph such that no $k$th power algorithm outputs a $(\frac{1}{\sqrt2}+\epsilon)$-approximation for computing the most robust matching.

In this article, we study an approximation algorithm for the maximum edge-disjoint paths problem. In this problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be connected by edge-disjoint paths. We give an O (log n )-approximation algorithm for the maximum edge-disjoint paths problem when an input graph is either 4-edge-connected planar or Eulerian planar. This improves an O (log 2 n )-approximation algorithm given by Kleinberg [2005] for Eulerian planar graphs. Our result also generalizes the result by Chekuri et al. [2004, 2005] who gave an O (log n )-approximation algorithm for the maximum edge-disjoint paths problem with congestion two when an input graph is planar.

The well-known Erdős-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c · k log k (for some constant c) such that G – X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdős-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G − X has no S-cycle, or conclude that such a vertex set does not exist in O(n2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n2m) time. We also extend the above results to those with the parity constraints as follows; 1. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G − X has no even S-cycle, or concludes that such a vertex set does not exist. 2. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G − X has no odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

A graph G contains a subdivision of H if G contains a subgraph which is isomorphic to a graph that can be obtained from H by subdividing some edges. A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. Although the well-known Kuratowski's theorem can be stated in terms of both a subdivision and a minor, we know that the notions of a subdivision and a minor do not seem to be similar. The notions of an immersion and a minor seem to be quite similar, and structural approach concerning graph minors has been extremely successful. In fact, Robertson and Seymour extended their proof of the famous Wanger's conjecture to prove that graphs are well-quasi-ordered by the immersion relation. We give additive approximation algorithms for list-coloring within 3.5(k + 1) of the list-chromatic number for graphs without Kk as a subdivision, and within 1.5(k − 1) of the list-chromatic number for graphs without Kk as an immersion. Clearly our results give rise to additive approximation algorithms for graph-coloring of graphs without Kk as a subdivision (in fact, we shall give an additive approximation algorithm within 2.5(k + 1) of the chromatic number) and Kk as an immersion, too. These are the first results in this direction (in fact, these are the first results concerning list-coloring graphs without fixed graph as a subdivision or as an immersion, except for the known upper bound results) and extend the result by Kawarabayashi, Demaine and Hajiaghayi (SODA'09) concerning the additive approximation algorithm for list-coloring graphs without Kk as a minor. We also discuss how our results are related to the famous Hájos’ conjecture and Hadwiger's conjecture. We point out that it is Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [6], and hence it is also Unique-Game hard to obtain an O(k/ log2 k)-approximation algorithm for graph-coloring of graphs without a Kk-subdivision or without a Kk-immersion. Therefore it really makes sense to consider an additive approximation algorithm for graph coloring of these family of graphs (which is in contrast to a 2-approximation algorithm for graph-coloring of H-minor-free graphs [13]).

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdös-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following: For any 4-edge-connected graph G=(V,E), either G has edge-disjoint k odd cycles or there exists an edge set F subseteq E with |F| <= f(k) such that G-F is bipartite. We note that the 4-edge-connectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edge-connected graph with n vertices. We show that, for any epsilon > 0, if k = O ((log log log n)^{1/2-epsilon}), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n.

A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner's Conjecture about the structure of minor-closed graph properties. In 2008, Demaine and Hajiaghayi proved a remarkable linear min-max relation for graphs excluding any fixed minor H: every H-minor-free graph of treewidth at least c_H r has an r times r-grid minor for some constant c_H. However, as they pointed out, there is still a major problem left in this theorem. The problem is that their proof heavily depends on Graph Minor Theory, most of which lacks explicit bounds and is believed to have very large bounds. Hence c_H is not explicitly given in the paper and therefore this result is usually not strong enough to derive efficient algorithms. Motivated by this problem, we give another (relatively short and simple) proof of this result without using big machinery of Graph Minor Theory. Hence we can give an explicit bound for c_H (an exponential function of a polynomial of |H|). Furthermore, our result gives a constant w=2^O(r^2 log r) such that every graph of treewidth at least w has an r times r-grid minor, which improves the previously known best bound 2^Theta(r^5)$ given by Robertson, Seymour, and Thomas in 1994.

Abstract This article describes a node capacitated in‐tree packing problem. The input consists of a directed graph, a root node, a node capacity function, and edge consumption functions. The problem is to find the maximum number of rooted in‐trees, such that the total consumption of in‐trees at each node does not exceed the capacity of the node. The problem is one of the network lifetime problems that are among the most important issues in the context of sensor networks. We establish the computational complexity of the problem under various restrictions on consumption functions and graphs. For example, we consider general graphs, acyclic graphs, and complete graphs embedded in the d ‐dimensional space \input amssym ${\Bbb{R}}^d$ having edge consumption functions depending only on distances between end nodes. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

A graph with at least $2k$ vertices is said to be k-linked if for any two ordered k-tuples $(s_1, \dots , s_k)$ and $(t_1, \dots , t_k)$ of $2k$ distinct vertices, there exist pairwise vertex-disjoint paths $P_1, \dots , P_k$ such that $P_i$ connects $s_i$ and $t_i$ for $i=1, \dots , k$. For a given graph G, we consider the problem of finding a maximum induced subgraph of G that is not k-linked. This problem is a common generalization of computing vertex-connectivity and testing k-linkedness of G, and it is closely related to the concept of H-linkedness. In this paper, we give the first polynomial-time algorithm for the case of $k=2$, whereas a similar problem to find a maximum induced subgraph without 2-vertex-disjoint paths connecting fixed terminal pairs is NP-hard. For the case of general k, we give an $(8k-2)$-additive approximation algorithm. We also investigate the computational complexity of the edge-disjoint case and the directed case.

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r-approximation algorithm for this problem is a polynomial time algorithm that finds at least OPT / r edge-disjoint paths, where OPT is the maximum possible. Currently, an O(n1/2)-approximation algorithm is best known for this problem even if a congestion of two is allowed, i.e., each edge is allowed to be used in at most two of the paths.

In this paper, we consider the half-integral disjoint paths packing. For a graph [Formula: see text] and [Formula: see text] pairs of vertices [Formula: see text] in [Formula: see text], the objective is to find paths [Formula: see text] in [Formula: see text] such that [Formula: see text] joins [Formula: see text] and [Formula: see text] for [Formula: see text], and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with [Formula: see text]. This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530–539] who proved the same conclusion when [Formula: see text]. Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a “crossbar” of a polynomial size of the tree width of the graph.

We consider the following well-known problem, which is called the edge-disjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2), …, (sk, tk) in G. Output: Edge-disjoint paths P1, P2, …, Pk in G such that Pi joins si and ti for i = 1, 2, …, k. Robertson and Seymour's graph minor project gives rise to an O(m3) algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project, spanning 23 papers and at least 500 pages proof. We give a faster algorithm and a simpler proof of the correctness for the edge-disjoint paths problem for any fixed k. Our results can be summarized as follows: 1. If an input graph G is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤ 3-edge-cuts. (iii) Excluding large clique minors. 2. When an input graph G is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be non-trivially superconstant number, up to k = O((log log log n)½–ε) for any ε > 0. Thus our hidden constant in this case is dramatically smaller than Robertson-Seymour's. In addition, if an input graph G is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((log n)½–ε) for any ε > 0. The same thing holds for bounded genus graphs. Moreover, if an input graph is either 4-edge-connected H-minor-free or Eulerian H-minor-free for fixed graph H, k is allowed to be O((log log n)½–ε) for any ε > 0. 3. We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, the time complexity of our algorithm is O(n2), which is faster than Robertson and Seymour's.

In this paper, we consider the problem of finding an induced cycle passing through k given vertices, which we call the induced cycle problem. The significance of finding induced cycles stems from the fact that precise characterization of perfect graphs would require structures of graphs without an odd induced cycle, and its complement. There has been huge progress in the recent years, especially, the Strong Perfect Graph Conjecture was solved in [6]. Concerning recognition of perfect graphs, there had been a long-standing open problem for detecting an odd hole and its complement, and finally this was solved in [4]. Unfortunately, the problem of finding an induced cycle passing through two given vertices is NP-complete in a general graph [2]. However, if the input graph is constrained to be planar and k is fixed, then the induced cycle problem can be solved in polynomial time [13, 14, 16]. In particular, an O(n2) time algorithm is given for the case k = 2 by McDiarmid, Reed, Schrijver and Shepherd [18], where n is the number of vertices of the input graph. Our main results in this paper are to improve their result in the following sense. 1. The number of vertices k is allowed to be non-trivially super constant number, up to . More precisely, when , then the ICP in planar graphs can be solved in O(n2+∊) time for any ∊ > 0. 2. The time complexity is linear if the given graph is planar and k is fixed. 3. The above results are extended to graphs embedded in a fixed surface. We note that the linear time algorithm (the second result) is independent from the first result. Let us point out that we give the first polynomial time algorithm for the problem for the bounded genus case. In fact, our proof gives a short proof of a result announced in [20] (without complete proof) which gives a linear time algorithm for the disjoint paths problem for fixed k for the bounded genus case. We also extend this result to the induced disjoint paths problem. Let us observe that if k is as a part of the input, then the problem is still NP-complete, and so we need to impose some condition on k.

A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta‐matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M‐convex functions on constant‐parity jump systems was introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M‐convex functions on constant‐parity jump systems generalize valuated matroids, valuated delta‐matroids, and M‐convex functions on base polyhedra. This paper reveals that the class of M‐convex functions on constant‐parity jump systems is closed under a number of natural operations such as splitting, aggregation, convolution, composition, and transformation by networks. The present results generalize hitherto‐known similar constructions for matroids, delta‐matroids, valuated matroids, valuated delta‐matroids, and M‐convex functions on base polyhedra.