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1 Spanning Circuits in Reguar Matroids Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see Fedor V. Fomin Petr A. Goovach Danie Lokshtanov Saket Saurabh Abstract We consider the fundamenta Matroid Theory probem of finding a circuit in a matroid spanning a set T of given termina eements. For graphic matroids this corresponds to the probem of finding a simpe cyce passing through a set of given termina edges in a graph. The agorithmic study of the probem on reguar matroids, a supercass of graphic matroids, was initiated by Gavenčiak, Krá, and Oum [ICALP 12], who proved that the case of the probem with T = 2 is fixed-parameter tractabe (FPT) when parameterized by the ength of the circuit. We extend the resut of Gavenčiak, Krá, and Oum by showing that for reguar matroids the Minimum Spanning Circuit probem, deciding whether there is a circuit with at most eements containing T, is FPT parameterized by k = T ; the Spanning Circuit probem, deciding whether there is a circuit containing T, is FPT parameterized by T. We note that extending our agorithmic findings to binary matroids, a supercass of reguar matroids, is highy unikey: Minimum Spanning Circuit parameterized by is W[1]- hard on binary matroids even when T = 1. We aso show a imit to how far our resuts can be strengthened by considering a smaer parameter. More precisey, we prove that Minimum Spanning Circuit parameterized by T is W[1]-hard even on cographic matroids, a proper subcass of reguar matroids. 1 Introduction Deciding if a given graph G contains a cyce passing through a specified set T of termina edges or vertices is the cassica probem in graph theory. The study of this probem can be traced back to the fundamenta theorem of Dirac from 1960s about the existence of a cyce in k-connected graph passing through a given set of k vertices [9]. According to Kawarabayashi [17]...cyces through a vertex set or an edge set are one of centra topics in a of graph theory. We refer to [16] for an overview on the graph-theoretica study of the probem, incuding the famous Lovász-Wooda Conjecture. The agorithmic version of this question, is there a poynomia time agorithm deciding if a given graph The research eading to these resuts has received funding from the European Research Counci under the European Union s Seventh Framework Programme (FP/ ) / ERC Grant Agreement n and the Research Counci of Norway via the project CLASSIS Department of Informatics, University of Bergen, Norway. Institute of Mathematica Sciences, Chennai, India contains a cyce passing through the set of termina vertices or edges, is the probem of a fundamenta importance in graph agorithms. Since the probem generaizes the cassica Hamitonian cyce probem, it is NP-compete. However, for a fixed number of terminas the probem is sovabe in poynomia time. The case T = 1 with one termina vertex or edge is triviay soved by the breadth first search. The case of T = 2 can be reduced to finding a fow of size 2 between two vertices in a graph. The case of T = 3 is aready nontrivia and was shown to be sovabe in inear time in [20], see aso [12]. The fundamenta resut of Robertson and Seymour on the disjoint path probem [24] impies that the probem can be soved in poynomia time for a fixed number of terminas. Kawarabayashi in [17] provided a quantitative improvement by showing that the probem is sovabe in poynomia time for T = O((og og n) 1/10 ), where n is the size of the input graph. Björkund et a. [2] gave a randomized agorithm soving the probem in time 2 T n O(1). The agorithm of Björkund et a. soves aso the minimization variant of the probem, where the task is to find a cyce of minimum ength passing through termina vertices. We refer to the book of Cygan et a. [5] for an overview of different techniques in parameterized agorithms for soving probems about cyces and paths in graphs. Matroids are combinatoria objects generaizing graphs and inear independence. The study of circuits containing certain eements of a matroid is one of the centra themes in matroid theory. For graphic matroids, the probem of finding a circuit spanning (or containing) a given set of eements corresponds to finding in a graph a simpe cyce passing through specified edges. The cassica theorem of Whitney [30] asserts that any pair of eements of a connected matroid are in a circuit. Seymour [27] obtained a characterization of binary matroids with a circuit containing a tripe of eements. See aso [6, 21, 23] and references there for combinatoria resuts about circuits spanning certain eements in matroids. However, compared to graphs, the agorithmic aspects of circuits through eements in matroids are much ess understood. In their work on deciding first order properties on matroids of ocay bounded branch-width, Gavenčiak et a. [14] initiated the agorithmic study of the foowing 1433 Copyright by SIAM Unauthorized reproduction of this artice is prohibited

2 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see probem. Minimum Spanning Circuit Input: Task: A binary matroid M with a ground set E, a weight function w : E N, a set of terminas T E, and a nonnegative integer. Decide whether there is a circuit C of M with w(c) such that T C. Here and further we assume that the set of natura numbers N = {1, 2,...}, that is, it does not incude 0. Since graphic matroids are binary, this probem is a generaization of the probem of finding a cyce through a given set of edges in a graph. By the resut of Vardy [29] about the Minimum Distance probem from coding theory, Minimum Spanning Circuit is NP-compete even when T =. Gavenčiak et a. [14] observed that the hardness resut of Downey et a. from [11] aso impies that Minimum Spanning Circuit is W[1]-hard on binary matroids with unitweights eements when parameterized by even if T = 1. Parameterized compexity of Minimum Spanning Circuit for T = on binary matroids, i.e. the case when we ask about the existence of a circuit of ength at most, is known as Even Set in parameterized compexity and is a ong standing open probem in the area. The intractabiity of the probem changes when we restrict the input binary matroid to be reguar, i.e. matroid which has a representation by rows of a totay unimoduar matrix. In particuar, Gavenčiak et a. show that for T = 2, Minimum Spanning Circuit is fixed parameter tractabe (FPT) being parameterized by by giving time O() n O(1) agorithm, where n is the number of eements in the input matroid. Reca that a graphic and cographic matroids are reguar and thus agorithmic resuts for reguar matroids yied agorithms on graphic and cographic matroids. Our resuts. In this work we show, and this is the main resut of the paper, that on reguar matroids Minimum Spanning Circuit is FPT being parameterized by without any additiona condition on the size of the termina set. Actuay, we obtain the agorithm for stronger parameterization k = w(t ). The running time of our agorithm is 2 O(k2 og k) n O(1). Our approach is based on the cassica decomposition theorem of Seymour [26]. Roughy speaking, the theorem aows to decompose a reguar matroid by making use of 1,2, and 3-sums into graphic, cographic matroids and matroid of a fixed size. (We refer to Section 3 for the precise formuation of the theorem). Thus to sove the probem on reguar matroids, one has to understand how to sove a certain extension of the probem on graphic and cographic matroids (matroids of constant size are usuay trivia), and then empoy Seymour s theorem to combine soutions. This is exacty the approach which was taken by Gavenčiak et a. in [14] for soving the probem for T = 2, and this is the approach we adapt in this paper. However, the detais are very different. In particuar, in order to use the Seymour s decomposition, we have to sove the probem on cographic matroids, which is aready quite non-obvious. Gavenčiak et a. [14] adapt the method of Kawarabayashi and Thorup [19] who used it to prove that finding an edge-cut with at most s edges that separates the input graph into at east k component is FPT when parameterized by s. This approach works for T = 2 and probaby may be extended for the case when the number of terminas is bounded, but we doubt that it coud be appied for the parameterization by k = w(t ). Hence, in order to sove Minimum Spanning Circuit on cographic matroids, we use the recent framework of recursive understanding deveoped by Chitnis et a. in [3] for the Minima Termina Cut probem. In this probem, we are given a a connected graph G with a termina set of edges T E(G) and termina vertex sets R 1, R 2 V (G), and the task is to find a cut C of sma weight satisfying a number of constraints: (a) this cut shoud be a minima cut-set, (b) it shoud contain a edges of T, and (c) it shoud separate R 1 from R 2, meaning that G C contains distinct connected components X 1 and X 2 such that R i X i for i {1, 2}. We beieve that this probem is interesting on its own. Finay, constructing a soution by going through Seymour s matroid decomposition when T is unbounded is aso a non-trivia procedure requiring a carefu anayses. With a simiar approach, we aso obtain an agorithm for the foowing decision version of the probem, where we put no constraints on the size of the circuit. Spanning Circuit Input: Task: A binary matroid M with a ground set E and a set of terminas T E. Decide whether there is a circuit C of M such that T C. We show that on reguar matroids Spanning Circuit is FPT parameterized by T. Due to space imitations, we ony sketch our agorithm for Minimum Spanning Circuit on reguar matroids and omit the proof of our resut for Spanning Circuit. The fu version of the ppaer with compete proofs is avaiabe at [13] Copyright by SIAM Unauthorized reproduction of this artice is prohibited

3 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see 2 Preiminaries Graphs. We consider finite undirected (muti) graphs that can have oops or mutipe edges. We refer to the book of Dieste [7] for the standard graph theory definitions and notations. Throughout the paper we use n to denote the number of vertices and m the number of edges of considered graphs uness it creates confusion. A set S E(G) is an (edge) cut-set if the deetion of S increases the number of components. A cut-set S is (incusion) minima if any proper subset of S is not a cut-set. Matroids. We refer to the book of Oxey [22] for the detaied introduction to matroid theory. We denote the ground set of a matroid M by E(M) and the set of independent set by I(M) or simpy by E and I if it does not creates confusion. If a set X E is not independent, then X is dependent. An (incusion) minima dependent set is caed a circuit of M. We denote the set of a circuits of M by C(M) or simpy C if it does not create a confusion. A circuit of the dua matroid M is caed cocircuit of M. An one-eement circuit is caed oop and an one-eement cocircuit is caed a cooop. If {e 1, e 2 } is a two-eement circuit, then it is said that e 1 and e 2 are parae. A set X E is a cyce of M if X either empty or X is a disjoint union of circuits. Matroids associated with graphs. Let G be a graph. The cyce matroid M(G) has the ground set E(G) and a set X E(G) is independent if X = or G[X] has no cyces. Notice that C is a circuit of M(G) if and ony if C induces a cyce of G. The bond matroid M (G) with the ground set E(G) is dua to M(G), and X is a circuit of M (G) if and ony if X is a minima cutset of G. Respectivey, Minimum Spanning Circuit for a cyce matroid M(G) is to decide whether G has a cyce C of weight at most that goes through the edges of T, and for a bond matroid M (G) it is to decide whether G has a minima cut-set C of weight at most that contains T. We say that M is a graphic matroid if M is isomorphic to M(G) for some graph G. Respectivey, M is cographic if there is graph G such that M is isomorphic to M (G). Matroid representations. Let M be a matroid and et F be a fied. An n m-matrix A over F is a representation of M over F if there is one-to-one correspondence f between E and the set of coumns of A such that for any X E, X I if and ony if the coumns f(x) are ineary independent (as vectors of F n ); if M has such a representation, then it is said that M has a representation over F. In other words, A is a representation of M if M is isomorphic to the coumn matroid of A, i.e. the matroid whose ground set is the set of coumns of A and a set of coumns is independent if and ony if these coumns are ineary independent. A matroid is binary if it can be represented over GF(2). A matroid is reguar if it can be represented over any fied. In particuar, graphic and cographic matroids are reguar. As we are working with binary matroids, we assume that for an input matroid, we are given its representation over GF(2). Then it can be checked in poynomia time whether a subset of the ground set is independent by checking the inear independence of the corresponding coumns. 3 Structure of reguar matroids Our resuts for reguar matroids use the structura decomposition for reguar matroids given by Seymour [25]. To describe the decomposition of matroids we need the notion of r-sums of matroids. However for our purpose it is sufficient that we restrict ourseves to binary matroids and up to 3-sums. We refer to [28, Chapter 8] for a more detaied introduction to matroid sums. Reca that, for two set X and Y, X Y = (X \ Y ) (Y \ X) denotes the symmetric difference of X and Y. Let M 1 and M 2 be binary matroids. The sum of M 1 and M 2, denoted by M 1 M 2, is the matroid M with the ground set E(M 1 ) E(M 2 ). The cyces of M are a subsets C E(M 1 ) E(M 2 ) of the form C 1 C 2, where C 1 is a cyce of M 1 and C 2 is a cyce of M 2. This does indeed define a binary matroid [25] in which the circuits are the minima non-empty cyces and the independent sets are (as aways) the sets that do not contain any circuit. For our purpose the foowing specia cases of matroid sums are sufficient. 1. If E(M 1 ) E(M 2 ) = and E(M 1 ), E(M 2 ), then M is the 1-sum of M 1 and M 2 and we write M = M 1 1 M If E(M 1 ) E(M 2 ) = 1, the unique e E(M 1 ) E(M 2 ) is not a oop or cooop of M 1 or M 2, and E(M 1 ), E(M 2 ) 3, then M is the 2-sum of M 1 and M 2 and we write M = M 1 2 M If E(M 1 ) E(M 2 ) = 3, the 3-eement set Z = E(M 1 ) E(M 2 ) is a circuit of M 1 and M 2, Z does not contain a cocircuit of M 1 or M 2, and E(M 1 ), E(M 2 ) 7, then M is the 3-sum of M 1 and M 2 and we write M = M 1 3 M 2. If M = M 1 r M 2 for some r {1, 2, 3}, then we write M = M 1 M 2. Definition 3.1. A {1, 2, 3}-decomposition of a matroid M is a coection of matroids M, caed the basic matroids and a rooted binary tree T in which M is the root and the eements of M are the eaves such that any interna node is either 1-, 2- or 3-sum of its chidren Copyright by SIAM Unauthorized reproduction of this artice is prohibited

4 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see We aso need the specia binary matroid R 10 to be abe to define the decomposition theorem for reguar matroids. It is represented over GF(2) by the matrix whose coumns are formed by vectors that have exacty three non-zero entries (or rather three ones) and no two coumns are identica. The foowing decomposition theorem for reguar matroids is due to Seymour [25]. Theorem 3.1. ([25]) Every reguar matroid M has an {1, 2, 3}-decomposition in which every basic matroid is either graphic, cographic, or isomorphic to R 10. Moreover, such a decomposition (together with the graphs whose cyce and bond matroids are isomorphic to the corresponding basic graphic and cographic matroids) can be found in time poynomia in E(M). For our agorithmic purposes we wi not use the Theorem 3.1 but rather a modification proved by Dinitz and Kortsarz in [8]. Dinitz and Kortsarz in [8] observed that some restrictions in the definitions of 2- and 3- sums are not important for the agorithmic purposes. In particuar, in the definition of the 2-sum, the unique e E(M 1 ) E(M 2 ) is not a oop or cooop of M 1 or M 2, and E(M 1 ), E(M 2 ) 3 coud be dropped. Simiary, in the definition of 3-sum the conditions that Z = E(M 1 ) E(M 2 ) does not contain a cocircuit of M 1 or M 2, and E(M 1 ), E(M 2 ) 7 coud be dropped. We define extended 1-, 2- and 3-sums by omitting these restrictions. Ceary, Theorem 3.1 hods if we repace sums by extended sums in the definition of the {1, 2, 3}-decomposition. To simpify notations, we use 1, 2, 3 and to denote these extended sums. Finay, we aso need the notion of a confict graph associated with a {1, 2, 3}-decomposition of a matroid M given by Dinitz and Kortsarz in [8]. Definition 3.2. ([8]) Let (T, M) be a {1, 2, 3}- decomposition of a matroid M. The intersection (or confict) graph of (T, M) is the graph G T with the vertex set M such that distinct M 1, M 2 M are adjacent in G T if and ony if E(M 1 ) E(M 2 ). Dinitz and Kortsarz in [8] showed how to modify a given decomposition in order to make the confict graph a forest. In fact they proved a sighty stronger condition that for any 3-sum (which by definition is summed aong a circuit of size 3), the circuit in the intersection is contained entirey in two of the owest-eve matroids. In other words, whie the process of summing matroids might create new circuits that contain eements that started out in different matroids, any circuit that is used as the intersection of a sum existed from the very beginning. Theorem 3.2. ([8]) For a given reguar matroid M, there is a (confict) tree T, whose set of nodes is a set of matroids M, where each eement of M is a graphic or cographic matroid, or a matroid obtained from R 10 by (possibe) deeting some eements and adding parae eements, that has the foowing properties: i) if two distinct matroids M 1, M 2 M have nonempty intersection, then M 1 and M 2 are adjacent in T, ii) for any distinct M 1, M 2 M, E(M 1 ) E(M 2 ) = 0, 1 or 3, iii) M is obtained by the consecutive performing extended 1, 2 or 3-sums for adjacent matroids in any order. Moreover, T can be constructed in a poynomia time. For a confict tree T of a matroid M, we say that M is defined by T. In our agorithms we work with rooted confict trees. Thus for the nodes of the rooted tree T, we have naturay defined parent-chid, descendant and ancestor reationships. Our agorithms are based on performing bottom-up traversa of the tree T. We say that a node M of T is a eaf if it has no chidren, and M s is a sub-eaf if it has at east one chid and the chidren of M s are eaves. Let M be a eaf and et M s be its adjacent sub-eaf. We say that M is an h-eaf for h {1, 2, 3} if the edge between M s and M corresponds to the extended h-sum. Since in Minimum Spanning Circuit and Spanning Circuit we are ooking for circuits containing termina, we need the foowing emma characterizing the structure of circuits of matroid sums. Lemma 3.1. Let M = M 1 r M 2 for r {1, 2, 3}, where M 1 and M 2 are binary matroids, and Z = E(M 1 ) E(M 2 ). i) If r = 1, then C(M) = C(M 1 ) C(M 2 ). ii) If r = 2 and Z = {e}, then C(M) ={C C(M 1 ) e / C} {C C(M 2 ) e / C} {C 1 C 2 C 1 C(M 1 ), C 2 C(M 2 ), e C 1, e C 2 } Copyright by SIAM Unauthorized reproduction of this artice is prohibited

5 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see iii) If r = 3, then C(M) ={C C(M 1 ) Z C = } {C C(M 2 ) Z C = } {C 1 C 2 C 1 C(M 1 ), C 2 C(M 2 ), C 1 Z = {e} and C 2 Z = {e} for some e Z, and C 1 Z C(M 1 ) or C 2 Z C(M 2 )}. 4 Minima cut with specified edges To construct an agorithm for Minimum Spanning Circuit for reguar matroids, we need an agorithm for cographic matroids. Let G be a connected graph, and et T E(G) be a set of termina edges. For sets R 1, R 2 V (G), we say that C E(G) is (R 1, R 2 )- termina cut-set if C is (a) a minima cut-set; (b) C T ; and (c) G C contains distinct connected components X 1 and X 2 such that R i X i for i {1, 2}. We wi need sove the foowing auxiiary parameterized probem Minima Termina Cut parameterized by k Input: A connected graph G, a weight function w : E(G) N, a set of terminas T E(G), sets R 1, R 2 V (G), and a positive integer k. Task: Decide whether G contains an (R 1, R 2 )- termina cut-set C such that w(c) w(t ) k. We say that an (R 1, R 2 )-termina cut-set C with the required weight is a soution of Minima Termina Cut. Observe that if for some instance of Minima Termina Cut we have that R 1 R 2, then this instance does not does not have a soution and thus is a no-instance. We show that Minima Termina Cut is FPT. In the specia case when R 1 = R 2 =, Minima Termina Cut essentiay asks for a minimum weight minima cut of a graph that contains specified edges. We beieve that this graph probem is interesting in its own. Theorem 4.1. Minima Termina Cut is sovabe in time 2 O(k2 og k) n O(1). The proof of Theorem 4.1 is based on a (nontrivia) appication of the recent agorithmic technique of recursive understanding introduced by Chitnis et a. in [3] (see aso [4] for more detais). 5 Soving Minimum Spanning Circuit on reguar matroids This section is devoted to the proof of the first main resut of the paper. Theorem 5.1. Minimum Spanning Circuit is sovabe in time 2 O(k2 og k) n O(1) on reguar n-eement matroids, where k = w(t ). The remaining part of the section contains a sketch the proof of the theorem. For technica reasons, in our agorithm we sove a specia variant of Minimum Spanning Circuit. In particuar, in our agorithm, the information about circuits in M wi be derived from circuits of size 3. We need the foowing technica definition. Definition 5.1. (Circuit constraints and extensions) Let M be a binary matroid given together with a set of terminas T E(M), and a famiy X of pairwise disjoint circuits of M of size 3, which are aso disjoint with T. Then a circuit constraint for M, T and X is an 8-tupe (M, T, X, P, Z, w, W, k), where P is a mapping assigning to each X X a nonempty set P (X) of subsets of X of size 1 or 2, Z is either the empty set, or is a pair of the form (Z, t), where Z is a circuit of size 3 disjoint with the circuits of X and with terminas T, and t is an eement of Z, w is a weight function, w : E \ L N, where L = X X X, W = {w X X X } is a famiy of weight functions, where w X : P (X) N for each X X, and k is an integer. We say that a circuit C of M is a feasibe extension satisfying circuit constraint (M, T, X, P, Z, w, W, k) (or just feasibe when it is cear from the context) if C X P (X) for each X X, if Z, then C Z is a circuit of M and Z C = {t}, and w(c \ (T L)) + X X w X(C X) k. This aows us to define the foowing auxiiary probem. Extended Minimum Circuit parameterized by k Input: A circuit constraint (M, T, X, P, Z, w, W, k). Task: Decide whether there is an extension satisfying the circuit constraint Copyright by SIAM Unauthorized reproduction of this artice is prohibited

6 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see Notice that Minimum Spanning Circuit parameterized by k = w(t ) is the specia case of Extended Minimum Circuit for X = and Z =. We ca a circuit C satisfying the requirements of the probem, i.e. which is an extension satisfying the corrsponding circuit constraint, by a soution. We aso refer to the vaue ω(c) = w(c \ (T L)) + X X w X(C X) as to the weight of C. In Section 5.1 we sove Extended Minimum Circuit on matroids of basic types, and in Section 5.2 we construct the agorithm for reguar matroids. 5.1 Soving Minimum Spanning Circuit on basic matroids For matroids obtained from R 10 by deeting eements and adding parae eements, the soution is easy. Lemma 5.1. Extended Minimum Circuit can be soved in poynomia time on the cass of matroids that can be obtained from R 10 by adding parae eements and deeting some eements. To construct an agorithm for Extended Minimum Circuit for graphic matroids, we consider the foowing parameterized probem: Cyce Through Terminas parameterized by k Input: Task: A graph G, a weight function w : E(G) N, a set of terminas T E(G), and a positive integer k. Does G has a cyce C with T E(C) such that w(e(c)) w(t ) k? This probem can be soved in time 2 k n O(1) by making use of the randomized agorithm of Björkund et a. [2]. As the running time of our agorithms for Minimum Spanning Circuit is dominated by the running time of the agorithm for cographic matroids, we give here deterministic agorithm with a worse constant in the base of the exponent. The agorithm is based of the coor coding technique of Aon et a. [1]. Lemma 5.2. Cyce Through Terminas is sovabe in time 2 O(k) n O(1). Appying this resut, we obtain the foowing emma. Lemma 5.3. Extended Minimum Circuit can be soved in time 2 O(k) E(M) O(1) on graphic matroids. To sove Extended Minimum Circuit on cographic matroids we empoy Theorem 4.1. Lemma 5.4. Extended Minimum Circuit can be soved in time 2 O(k2 og k) E(M) O(1) on cographic matroids. 5.2 Proof of Theorem 5.1 Now we are ready to give an agorithm for Minimum Spanning Circuit parameterized by k = w(t ) on reguar matroids. Let (M, w, T, ) be an instance of Minimum Spanning Circuit, where M is reguar. We consider it to be an instance (M, T, X, P, Z, w, W, k) of Extended Minimum Circuit, where X = and Z =. If M can be obtained from R 10 by the addition of parae eements or is graphic or cographic, we sove the probem directy using Lemmas Assume that it is not the case. Using Theorem 3.2, we find a confict tree T with a node set M. We seect a node r of T containing an eement of T as a root. We say that an instance (M, T, X, P, Z, w, W, k) of Extended Minimum Circuit is consistent (with respect to T ) if Z = and for any X X, X E(M ) for some M M. Ceary, the instance obtained from the origina input instance (M, w, T, ) of Minimum Spanning Circuit is consistent. We use reduction rues that remove eaves keeping this property. Let M M be a matroid that is a eaf of T. We construct reduction rues depending on whether M is 1, 2 or 3-eaf. Denote by M s its neighbor in T. Let aso T be the tree obtained from T be the deetion of M, and et M be the matroid defined by T. Ceary, M = M M. Throughout this section, we say that a reduction rue is safe if it either correcty soves the a probem or returns an equivaent instance of Extended Minimum Circuit together with corresponding confict tree of the obtained matroid that is consistent and the vaue of the parameter does not increase. The construction and the safeness proofs of the rues is based on the structura Lemma 3.1. From now, et (M, T, X, P, Z, w, W, k) be a consistent instance of Extended Minimum Circuit. note L = X X X. Reduction Rue 5.1. (1-Leaf reduction rue) If M is a 1-eaf, then do the foowing. De- i) If T E(M ) or there is X X such that X E(M ), then stop and return a no-answer, ii) Otherwise, return the instance (M, T, X, P,, w, W, k), where w is the restriction of w on E(M ) \ L, and sove it using the confict tree T. Since the root matroid contains at east one termina, Lemma 3.1 i) immediatey impies the foowing emma. Lemma 5.5. Reduction Rue 5.1 is safe and can be impemented in poynomia (in E(M) ) time Copyright by SIAM Unauthorized reproduction of this artice is prohibited

7 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see Reduction Rue 5.2. (2-Leaf reduction rue) If M is a 2-eaf, then et {e} = E(M ) E(M s ) and do the foowing. i) If T E(M ) = and there is no X X such that X E(M ), then find a circuit C of M containing e with minimum w(c \ {e}) k. If there is no such a circuit, then set w (e) = k + 1, and et w (e) = w(c \ {e} otherwise. Assume that w (e ) = w(e ) for e E(M ) \ L. Return the instance (M, T, X, P,, w, W, k) and sove it using the confict tree T. ii) Otherwise, if T E(M ) or there is X X such that X E(M ), consider T = (T E(M )) {e} and X = {X X X E(M )}. Define P, w, W by restricting the corresponding functions by E(M ) assuming additionay that w (e) = 1. Find the minimum k k such that (M, T, X, P,, w, W, k ) is a yes-instance of Extended Minimum Circuit. If such k does not exist, we concude that the given instance is a no-instance and stop. Otherwise, we do the foowing. Set T = (T E(M )) {e} and X = {X X X E(M )}. Define P, w, W by restricting the corresponding functions by E(M ) assuming additionay that w (e) = 1. Return the instance (M, T, X, P,, w, W, k k ) and sove it using the confict tree T. Lemma 5.6. Reduction Rue 5.2 is safe and can be impemented in time 2 O(k2 og k) E(M) O(1). Reduction Rue 5.3. (3-Leaf reduction rue) If M is a 3-eaf, then et S = {e 1, e 2, e 3 } = E(M ) E(M s ) and do the foowing. i) If T E(M ) = and there is no X X such that X E(M ), then for each i {1, 2, 3}, find a circuit C (i) of M such that C (i) S = {e i } and C (i) S is a circuit of M with minimum w(c (i) \ {e i }) k. If there is no such a circuit, then set w (e i ) = k + 1, and et w (e i ) = w(c (i) \ {e i }) otherwise. Assume that w (e ) = w(e ) for e E(M ) \ (L S). Return the instance (M, T, X, P,, w, W, k) and sove it using the confict tree T. ii) If there is no X X such that X E(M ), but T = T E(M ) and there is i {1, 2, 3} such that C = T {e i } is a circuit of M, then consider two cases. C S is a circuit of M. Set w (e i ) = 1 and assume that w (e ) = w(e ) for e E(M ) \ (S L). For each j {1, 2, 3} \ {i}, do the foowing. Let h {1, 2, 3} \ {i, j}. Set X = {S}, P (S) = {e j }, w S ({e h }) = 1 and W = {w S }. Let w be a restriction of w on E(M ). Find a minimum k (h) k + 1 such that (M, T, X, P,, w, W, k (h) ) is a yesinstance of Extended Minimum Circuit. If there is no such k (h), then set w (e j ) = k+1 and set w (e j ) = k (h) 1 otherwise. Set T = (T E(M )) {e i }. Return the instance (M, T, X, P,, w, W, k) and sove it using the confict tree T. C S is not a circuit of M. Set w (e i ) = k+1 and w (e j ) = 1 for j {1, 2, 3} \ {i}. Assume that w (e ) = w(e ) for e E(M ) \ (L S). Set T = (T E(M )) (S \ {e i }). Return the instance (M, T, X, P,, w, W, k) and sove it using the confict tree T. iii) Otherwise, et T = T E(M ) and X = {X X X E(M )}. Define P, w, W by restricting the corresponding functions by E(M ). Construct the set Y of subsets of S and the function w S : Y N as foows. Initiay, set Y =. Define w (e i) = 1 for i {1, 2, 3} and et w (e) = w (e) for e E(M ) \ (L S). For i {1, 2, 3}, find the minimum k (i) k + 1 such that (M, T, X, P, (S, e i ), w, W, k (i) ) is a yes-instance of Extended Minimum Circuit. If such k (i) exists, then add {e i } in Y and set w S ({e i }) = k (i) 1. Let X = X {S}. For each i {1, 2, 3}, do the foowing. Set P (i) (X) = P (X) for X X and P (i) w (i) S ({e i}) = 1 and W (i) Find the minimum k (i) that (M, T, X, P (i) (Y ) = {x i }, set = W {w (i) S }. k + 1 such,, w, W (i), k (i) ) is a yesinstance of Extended Minimum Circuit. If such k (i) exists, then add S \ {e i } in Y and set w S (S \ {e i }) = k (i) 1. If Y =, then we concude that we have a noinstance and stop. Otherwise, set T = T E(M ), X = {X X X E(M )} {S} and for X X, et P (X) = P (X) if X P (X) and P (S) = Y. Aso et W = {w X X X } and et w be the restriction of w on E(M ). Return the instance (M, T, X, P,, w, W, k) and sove it using the confict tree T Copyright by SIAM Unauthorized reproduction of this artice is prohibited

8 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see Lemma 5.7. Reduction Rue 5.3 is safe and can be appied in time 2 O(k2 og k) E(M) O(1). Now we can compete the proof of Theorem 5.1. Observe that M and the corresponding confict tree T can be constructed in poynomia time by Theorem 3.2, and then we appy the reduction rues at most V (T ) 1 times unti we obtain an instance of Extended Minimum Circuit for a matroid of one of basic types and sove the probem using Lemmas Lower bounds and open questions In this paper we gave FPT agorithms for Minimum Spanning Circuit and Spanning Circuit for reguar matroids. We concude with a number of open agorithmic questions about circuits in matroids. We aso discuss here certain agorithmic imitations for extending our resuts. Larger matroid casses. The first natura question is whether our resuts can be extended to other casses of matroids? There is no hope (of course up to certain compexity assumptions) that our resuts can be extended to binary matroids. Downey et a. proved in [11] that the foowing probem is W[1]-hard being parameterized by k. (We refer to the book of Downey and Feows [10] for the definition of W-hierarchy.) In the Maximum-Likeihood Decoding probem we are given a binary n m matrix A, a target binary n- eement vector s, and a positive integer k. The question is whether there is a set of at most k coumns of A that sum to s? As it was observed by Gavenciak et a. [14], the resut of Downey et a. immediatey impies the foowing proposition. Proposition 6.1. ([14]) Minimum Spanning Circuit is W[1]-hard on binary matroids with unit-weights eements when parameterized by even when T = 1. Let us note that Minimum Spanning Circuit with T = 0 on binary matroids is equivaent to Even Set, which parameterized compexity is a ong standing open question, see e.g. [10]. However Proposition 6.1 does not rue out a possibiity that our resuts can be extended from the cass of reguar matroids to any proper minor-cosed cass of binary, and even more generay, representabe over some finite fied, matroids. It is very ikey that the powerfu structura theorems obtained by Geeen et a. in order to sette Rota s conjecture, see [15] for further discussions, can shed some ight on this question. Soving both probems on transversa matroids is another interesting probem. Stronger parameterization. Björkund et a. in [2] gave a randomized agorithm that finds a shortest cyce through a given set T of vertices or edges in a graph in time 2 T n O(1). Hence Minimum Spanning Circuit parameterized by w(t ) is (randomized) FPT on graphic matroids if the weights are encoded in unary. Unfortunatey, it is possibe to show that Minimum Spanning Circuit is W[1]-hard aready on cographic matroids for this parameterization. Theorem 6.1. Minimum Spanning Circuit is W[1]- hard on cographic matroids with unit-weights eements when parameterized by T. Interestingy, Theorem 6.1 does not rue out a possibiity that for a fixed numbers of terminas Minimum Spanning Circuit is sti resovabe in poynomia time, or in other words that it is in XP parameterized by T. We conjecture that this is not the case. More precisey, is Minimum Spanning Circuit NPcompete on cographic matroids for a fixed number, say T = 3, termina eements? Other circuit probems. We do not know if our technique coud be adapted to sove the foowing variant of the spanning circuit probem. Given a reguar matroid M with a set of terminas, decide whether M contains a circuit of size at east spanning a terminas. We eave the compexity of this probem parameterized by open. Another interesting variation of Minimum Spanning Circuit and Spanning Circuit is the probem where we seek for a circuit of a given parity containing a given set of termina eements T. For graphs (or graphic matroids), Kawarabayshi et a. [18] proved that the probem is FPT parameterized by T. The compexity of this probem on cographic matroids is open. References [1] N. Aon, R. Yuster, and U. Zwick, Coor-coding, J. ACM, 42 (1995), pp [2] A. Björkund, T. Husfedt, and N. Tasaman, Shortest cyce through specified eements, in Proceedings of the 22nd Annua ACM-SIAM Symposium on Discrete Agorithms (SODA), SIAM, 2012, pp [3] R. H. Chitnis, M. Cygan, M. Hajiaghayi, M. Piipczuk, and M. Piipczuk, Designing FPT agorithms for cut probems using randomized contractions, in FOCS 2012, IEEE Computer Society, 2012, pp [4] R. H. Chitnis, M. Cygan, M. Hajiaghayi, M. Piipczuk, and M. Piipczuk, Designing FPT agorithms for cut probems using randomized contractions, CoRR, abs/ (2012) Copyright by SIAM Unauthorized reproduction of this artice is prohibited

9 Downoaded 11/28/17 to Redistribution subject to SIAM icense or copyright; see [5] M. Cygan, F. V. Fomin, L. Kowaik, D. Lokshtanov, D. Marx, M. Piipczuk, M. Piipczuk, and S. Saurabh, Parameterized Agorithms, Springer, [6] T. Deney and H. Wu, A generaization of a theorem of Dirac, J. Combinatoria Theory Ser. B, 82 (2001), pp [7] R. Dieste, Graph Theory, 4th Edition, vo. 173 of Graduate texts in mathematics, Springer, [8] M. Dinitz and G. Kortsarz, Matroid secretary for reguar and decomposabe matroids, SIAM J. Comput., 43 (2014), pp [9] G. A. Dirac, In abstrakten Graphen vorhandene voständige 4-Graphen und ihre Unterteiungen, Math. Nachr., 22 (1960), pp [10] R. G. Downey and M. R. Feows, Fundamentas of Parameterized Compexity, Texts in Computer Science, Springer, [11] R. G. Downey, M. R. Feows, A. Vardy, and G. Whitte, The parametrized compexity of some fundamenta probems in coding theory, SIAM J. Comput., 29 (1999), pp [12] H. Feischner and G. J. Woeginger, Detecting cyces through three fixed vertices in a graph, Inform. Process. Lett., 42 (1992), pp [13] F. V. Fomin, P. A. Goovach, D. Lokshtanov, and S. Saurabh, Spanning circuits in reguar matroids, CoRR, abs/ (2016). [14] T. Gavenciak, D. Krá, and S. Oum, Deciding first order properties of matroids, in Proceedings of the 39th Internationa Cooquium of Automata, Languages and Programming (ICALP), vo of Lecture Notes in Comput. Sci., Springer, 2012, pp [15] J. Geeen, B. Gerards, and G. Whitte, Soving Rota s conjecture, Notices Amer. Math. Soc., 61 (2014), pp [16] K. Kawarabayashi, One or two disjoint circuits cover independent edges: Lovász-wooda conjecture, J. Comb. Theory, Ser. B, 84 (2002), pp [17] K. Kawarabayashi, An improved agorithm for finding cyces through eements, in IPCO 2008, vo of Lecture Notes in Computer Science, Springer, 2008, pp [18] K. Kawarabayashi, Z. Li, and B. A. Reed, Recognizing a totay odd K 4-subdivision, parity 2-disjoint rooted paths and a parity cyce through specified eements, in Proceedings of the 20th Annua ACM-SIAM Symposium on Discrete Agorithms (SODA), SIAM, 2010, pp [19] K. Kawarabayashi and M. Thorup, The minimum k-way cut of bounded size is fixed-parameter tractabe, in Proceedings of the 52nd Annua Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2011, pp [20] A. S. LaPaugh and R. L. Rivest, The subgraph homeomorphism probem, J. Comput. System Sci., 20 (1980), pp [21] S. McGuinness, Ore-type and Dirac-type theorems for matroids, J. Combinatoria Theory Ser. B, 99 (2009), pp [22] J. G. Oxey, Matroid theory, Oxford University Press, [23] J. G. Oxey, A matroid generaization of a resut of Dirac, Combinatorica, 17 (1997), pp [24] N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths probem, J. Combinatoria Theory Ser. B, 63 (1995), pp [25] P. D. Seymour, Decomposition of reguar matroids, J. Comb. Theory, Ser. B, 28 (1980), pp [26] P. D. Seymour, Recognizing graphic matroids, Combinatorica, 1 (1981), pp [27] P. D. Seymour, Tripes in matroid circuits, European J. Combin., 7 (1986), pp [28] K. Truemper, Matroid decomposition, Academic Press, [29] A. Vardy, Agorithmic compexity in coding theory and the minimum distance probem, in Proceedings of the 29th Annua ACM Symposium on Theory of Computing (STOC), ACM, 1997, pp [30] H. Whitney, On the Abstract Properties of Linear Dependence, Amer. J. Math., 57 (1935), pp Copyright by SIAM Unauthorized reproduction of this artice is prohibited