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FixedParameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
"... Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a se ..."
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Cited by 14 (5 self)
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Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a set of at most p edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the multicut problem, in which we want to disconnect only a set of k given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixedparameter tractable (FPT) parameterized by p. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]hard parameterized by p. We complete the picture here by our main result which is that both DIRECTED VERTEX MULTIWAY CUT and DIRECTED EDGE MULTIWAY CUT can be solved in time 22O(p) nO(1) , i.e., FPT parameterized by size p of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that DIRECTED MULTICUT is FPT for the case of k = 2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011). 1
An 8Approximation Algorithm for the Subset Feedback Vertex Set Problem
 Problem, 37th Symp. on Foundations of Comp. Sci. (FOCS
, 1996
"... We present an 8approximation algorithm for the problem of finding a minimum weight subset feedback vertex set. The input in this problem consists of an undirected graph G = (V; E) with vertex weights w(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contai ..."
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We present an 8approximation algorithm for the problem of finding a minimum weight subset feedback vertex set. The input in this problem consists of an undirected graph G = (V; E) with vertex weights w(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset feedback vertex set with respect to S if it intersects every interesting cycle. The goal is to find a minimum weight subset feedback vertex set. The best pervious algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset feedback vertex set problem generalizes two NPComplete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multicommodity flow, which we call relaxed multicommodity flow. Relaxed multicommodity fl...
Greedy splitting algorithms for approximating multiway partition problems
 MATH. PROGRAMMING
, 2005
"... Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ·+f(Vk), ..."
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Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ·+f(Vk), where P is a kpartition of V if (i) Vi � = ∅, (ii) Vi ∩ Vj = ∅, i � = j, and (iii) V1 ∪ V2 ∪ · · · ∪ Vk = V hold. MPP formulation captures a generalization in submodular systems of many NPhard problems such as kway cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs.
Iterative compression for exactly solving nphard minimization problems
 in Algorithmics of Large and Complex Networks, Lecture Notes in Computer Science
"... Abstract. We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixedparameter algorithms for NPhard ..."
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Abstract. We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixedparameter algorithms for NPhard minimization problems. There is a clear potential for further applications as well as a further development of the technique itself. We describe several algorithmic results based on iterative compression and point out some challenges for future research. 1
New results on rationality and strongly polynomial solvability in eisenberggale markets
 In Proceedings of 2nd Workshop on Internet and Network Economics
, 2006
"... Abstract. We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a c ..."
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Abstract. We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatorial LP. This helps resolve positively the status of two markets left as open problems by [JV]: the capacity allocation market in a directed graph with two sourcesink pairs and the network coding market in a directed network with two sources. Our algorithms for solving the corresponding nonlinear convex programs are fundamentally different from those obtained by [JV]; whereas they use the primaldual schema, we use a carefully constructed binary search. 1
A New Approximation Algorithm for Finding Heavy Planar Subgraphs
 ALGORITHMICA
, 1997
"... We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NPHard problem of finding a heaviest planar subgraph in an edgeweighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM ..."
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Cited by 10 (3 self)
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We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NPHard problem of finding a heaviest planar subgraph in an edgeweighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that produces a maximum weight spanning tree in G. Based on the BermanRamaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1/3 + 1/72. We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3/8. Furthermore, we derive the first nontrivial performance ratio (7/12 instead of 1/2) for the NPHard MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem.
Cuts and disjoint paths in the valleyfree path model of Internet BGP routing
 IN COMBINATORIAL AND ALGORITHMIC ASPECTS OF NETWORKING
, 2004
"... In the valleyfree path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by BGP routing policies of autonomous systems in the Internet. Robustness considerations lead to the problem of c ..."
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In the valleyfree path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by BGP routing policies of autonomous systems in the Internet. Robustness considerations lead to the problem of computing a maximum number of disjoint paths between two nodes, and the minimum size of a cut that separates them. We study these problems in the valleyfree path model. For the problem of computing a maximum number of edge or vertexdisjoint valid paths between two given vertices s and t, we give a 2approximation algorithm and show that no better approximation ratio is possible unless P = NP. For the problem of computing a minimum vertex cut that separates s and t with respect to all valid paths, we give a 2approximation algorithm and prove that the problem is APXhard. The corresponding problem for edge cuts is shown to be polynomialtime solvable. We present additional results for acyclic graphs.
Instant Recognition of Half Integrality and 2Approximations
 In Proceedings of the 3rd International Workshop on Approximation Algorithms for Combinatorial Optimization
, 1998
"... . We define a class of integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax + by z + c, where the variable z appears only in that constraint. For such binary integer programs it is possible to derive half integral ..."
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. We define a class of integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax + by z + c, where the variable z appears only in that constraint. For such binary integer programs it is possible to derive half integral superoptimal solutions in polynomial time. The scheme is also applicable with few modifications to nonbinary integer problems. For some of these problems it is possible to round the half integral solution to a 2approximate solution. This extends the class of integer programs with at most two variables per constraint that were analyzed in [HMNT93]. The approximation algorithms here provide an improvement in running time and range of applicability compared to existing 2approximations. Furthermore, we conclude that problems in the framework are MAX SNPhard and at least as hard to approximate as vertex cover. Problems that are amenable to the analysis provided here are easily recognized. The ...
A Generalization of Totally Unimodular and Network Matrices
, 2002
"... In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these wellknown matrices. In particular, our focus is on the polyhedral consequences of totally unimodular ..."
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Cited by 7 (3 self)
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In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these wellknown matrices. In particular, our focus is on the polyhedral consequences of totally unimodular matrices, namely we look for matrices that can ensure vertices that are scalable to an integral vector by an integer k. We argue that simply generalizing the determinantal structure of totally unimodular matrices does not suffice to achieve this goal and one has to extend the range of values the inverses of submatrices can contain. To this end, we define kregular matrices. We show that kregularity is a proper generalization of total unimodularity in polyhedral terms, as it guarantees the scalability of vertices. Moreover, we prove that the kregularity of a matrix is necessary and sufficient for substituting modk cuts for rank1 ChvatalGomory cuts. In the second part of the thesis we introduce binet matrices, an extension of network matrices to bidirected graphs. We provide an algorithm to calculate the columns of a binet matrix using the underlying graphical structure. Using this method, we prove some results about binet matrices and demonstrate that several interesting classes of matrices are binet. We show that binet matrices are 2regular, therefore they provide halfintegral vertices for a polyhedron with a binet constraint matrix and integral right hand side vector. We also prove that optimization on such a polyhedron can be carried out very efficiently, as there exists an extension of the network simplex method for binet matrices. Furthermore, the integer optimization with binet matrices is equivalent to solving a matching problem. We also describe the connection o...