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Flexible network design deals with building a network that guarantees some connectivity requirements between its vertices, even when some of its elements (like vertices or edges) fail. In particular, the set of edges (resp. vertices) of a given graph are here partitioned into safe and unsafe. The goal is to identify a minimum size subgraph that is 2-edge-connected (resp. 2-vertex-connected), and stay so whenever any of the unsafe elements gets removed. In this paper, we provide improved approximation algorithms for flexible network design problems, considering both edge-connectivity and vertex-connectivity, as well as connectivity values higher than 2. For the vertex-connectivity variant, in particular, our algorithm is the first with approximation factor strictly better than 2.

The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are "short" if they have few steps or small total length. Both interpretations of short are relevant to the theory and application of linear programming. We study the hardness of several problems relating to the construction of short circuit walks. We establish that for a pair of vertices of a $0/1$-network-flow polytope, it is NP-complete to determine the length of a shortest circuit walk, even if we add the requirement that the walk must be sign-compatible. Our results also imply that determining the minimal number of circuits needed for a sign-compatible decomposition is NP-complete. Further, we show that it is NP-complete to determine the smallest total length (for $p$-norms $\lVert \cdot \rVert_p$, $1 < p \leq \infty$) of a circuit walk between a pair of vertices. One method to construct a short circuit walk is to pick up a correct facet at each step, which generalizes a non-revisiting walk. We prove that it is NP-complete to determine if there is a circuit direction that picks up a correct facet; in contrast, this problem can be solved in polynomial time for TU polyhedra.

The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point along an improving edge-direction of the underlying polyhedron. A key issue in the simplex algorithm's performance is degeneracy, which may lead to a (potentially long) sequence of basis exchanges which do not change the current extreme point solution. In this paper, we prove that it is always possible to limit the number of consecutive degenerate pivots that the simplex algorithm performs to $n-m-1$, where $n$ is the number of variables and $m$ is the number of equality constraints of a given LP in standard equality form., Comment: 18 pages, 2 figures

Capacitated network bargaining games are popular combinatorial games that involve the structure of matchings in graphs. We show that it is always possible to stabilize unit-weight instances of this problem (that is, ensure that they admit a stable outcome) via capacity-reduction and edge-removal operations, without decreasing the total value that the players can get. Furthermore, for general weighted instances, we show that computing a minimum amount of vertex-capacity to reduce to make an instance stable is a polynomial-time solvable problem. We then exploit this to give approximation results for the NP-hard problem of stabilizing a graph via edge-removal operations. Our work extends and generalizes previous results in the literature that dealt with a unit-capacity version of the problem, using several new arguments. In particular, while previous results mainly used combinatorial techniques, we here rely on polyhedral arguments and, more specifically, on the notion of circuits of a polytope., Comment: 28 pages

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs., Comment: 33 pages, 7 figures, submitted to IPCO 2023

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G. In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case. Finally, in unit-capacity graphs there is an equivalence between the stability of a graph, existence of a stable solution for network bargaining games, and existence of a stable solution for cooperative matching games. We show that this equivalence does not extend to the capacitated case., Comment: 14 pages, 3 figures

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1 polytopes and novel modifications to the classical Steepest-Edge and Shadow-Vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization., Comment: 32 Pages, 3 figures

Many network design problems deal with the design of low-cost networks that are resilient to the failure of their elements, such as nodes or links. One such problem is Connectivity Augmentation, where the goal is to cheaply increase the connectivity of a network from a value k to k+1. The most studied setting focuses on edge-connectivity, which reduces to k=2, called Cactus Augmentation. From an approximation perspective, Byrka, Grandoni, and Jabal Ameli (2020) were the first to break the 2-approximation barrier for this problem, by exploiting a connection to the Steiner Tree problem, and by tailoring the analysis of the iterative randomized rounding technique for Steiner Tree to the specific instances arising from this connection. Recently, Nutov (2020) observed that a similar reduction to Steiner Tree holds for a node-connectivity problem called Block-Tree Augmentation, where the goal is to add edges to a given spanning tree so that the resulting graph 2-node-connected. Combining Nutov's result with the algorithm of Byrka, Grandoni, and Jabal Ameli yields a 1.91-approximation for Block-Tree Augmentation, that is the best bound known so far. In this work, we give a 1.892-approximation algorithm for the problem of augmenting the node-connectivity of any graph from 1 to 2. As a corollary, we improve upon the state-of-the-art approximation factor for Block-Tree Augmentation. Our result is obtained by developing a different and simpler analysis of the iterative randomized rounding technique. Our results also imply a 1.892-approximation algorithm for Cactus Augmentation. While this does not beat the best approximation by Cecchetto, Traub, and Zenklusen (2021) known for this problem, our analysis is quite simple compared to previous results in the literature. In addition, our work gives new insights on the iterative randomized rounding method, that might be of independent interest.

The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their current partner in the matching. In this paper, we study the diameter of the polytope given by the convex hull of characteristic vectors of stable marriages, in the setting with ties. We prove an upper bound of $\lfloor \frac{n}{3}\rfloor$ on the diameter, where $n$ is the total number of men and women, and give a family of instances for which the bound holds tight. Our result generalizes the bound on the diameter of the standard stable marriage polytope (i.e., the well-known polytope that describes the setting without ties), developed previously in the literature.

Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called circuits, that is a superset of the edges of a polytope. We show that in the circuit-augmentation framework the greatest-improvement and Dantzig pivot rules are NP-hard, already for 0/1-LPs. Differently, the steepest-descent pivot rule can be carried out in polynomial time in the 0/1 setting, and the number of circuit augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1-LPs. The number of circuit augmentations has been of interest as a proxy for the number of steps in the Simplex method, and the circuit-diameter of polyhedra has been studied as a lower bound to the combinatorial diameter of polyhedra. Extending prior results, we show that for any polyhedron $P$ the circuit-diameter is bounded by a polynomial in the input bit-size of $P$. This is in contrast with the best bounds for the combinatorial diameter of polyhedra. Interestingly, we show that the circuit-augmentation framework can be exploited to make novel conclusions about the classical Simplex method itself: In particular, as a byproduct of our circuit results, we prove that (i) computing the shortest (monotone) path to an optimal solution on the 1-skeleton of a polytope is NP-hard, and hard to approximate within a factor better than 2, and (ii) for $0/1$ polytopes, a monotone path of strongly-polynomial length can be constructed using steepest improving edges.

Cooperative games are an important class of problems in game theory, where the goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial time characterization of submodular spanning tree games.

The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the \emph{fractional matching} polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an \emph{exact characterization} of the diameter of the fractional matching polytope, that is of independent interest.

The classical stable marriage problem asks for a matching between a set of men and a set of women with no blocking pairs, which are pairs formed by a man and a woman who would both prefer switching from their current status to be paired up together. When both men and women have strict preferences over the opposite group, all stable matchings have the same cardinality, and the famous Gale-Shapley algorithm can be used to find one. Differently, if we allow ties in the preference lists, finding a stable matching of maximum cardinality is an NP-hard problem, already when the ties are one-sided, that is, they appear only in the preferences of one group. For this reason, many researchers have focused on developing approximation algorithm for this problem. In this paper, we give a refined analysis of an approximation algorithm given by Huang and Telikepalli (IPCO14) for the stable marriage problem with one-sided ties, which shows an improved 13/9 -approximation factor for the problem. Interestingly, our analysis is tight.

The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined as the maximum value of a shortest path between two vertices of $P$, where the path uses potential edge directions of $P$ i.e., all edge directions that can arise by translating some of the facets of $P$. In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Salesman polytope and the Fractional Stable Set polytope.

An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of $G$. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from $G$ yields a stable graph, for any weighted graph $G$. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph $G$ yields a stable graph, does not admit any constant-factor approximation algorithm, unless $P=NP$. In this setting, we develop an $O(\Delta)$-approximation algorithm for the problem, where $\Delta$ is the maximum degree of a node in $G$.

We study a generalization of the Steiner tree problem, where we are given a weighted network $G$ together with a collection of $k$ subsets of its vertices and a root $r$. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for $k=2$, and give a $\frac{3}{2}$-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known $O(\log n)$-approximation. Despite these obstacles, we conjecture that this problem should have an $O(1)$-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree $G = (V,E)$, an additional set of edges $L$ called links and a cost vector $c \in \mathbb{R}^L_{\geq 1}$. The goal is to choose a minimum cost subset $S \subseteq L$ such that $G = (V, E \cup S)$ is $2$-edge-connected. In the unweighted case, that is, when we have $c_\ell = 1$ for all $\ell \in L$, the problem is called the tree augmentation problem (TAP). Both problems are known to be APX-hard, and the best known approximation factors are $2$ for WTAP by (Frederickson and J\'aJ\'a, '81) and $\tfrac{3}{2}$ for TAP due to (Kortsarz and Nutov, TALG '16). In the case where all link costs are bounded by a constant $M$, (Adjiashvili, SODA '17) recently gave a $\approx 1.96418+\varepsilon$-approximation algorithm for WTAP under this assumption. This is the first approximation with a better guarantee than $2$ that does not require restrictions on the structure of the tree or the links. In this paper, we improve Adjiashvili's approximation to a $\frac{3}{2}+\varepsilon$-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines $\{0,\frac{1}{2}\}$-Chv\'atal-Gomory cuts for the standard LP for the problem with bundle constraints from Adjiashvili. We show that our LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved guarantee of $\frac{3}{2}+\varepsilon$. For TAP, this is the best known LP-based result, and matches the bound of $\frac{3}{2}+\varepsilon$ achieved by the best SDP-based algorithm due to (Cheriyan and Gao, arXiv '15).

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(\delta(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+\epsilon)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).

We consider the classic problem of scheduling a set of n jobs non-preemptively on a single machine. Each job j has non-negative processing time, weight, and deadline, and a feasible schedule needs to be consistent with chain-like precedence constraints. The goal is to compute a feasible schedule that minimizes the sum of penalties of late jobs. Lenstra and Rinnoy Kan [Annals of Disc. Math., 1977] in their seminal work introduced this problem and showed that it is strongly NP-hard, even when all processing times and weights are 1. We study the approximability of the problem and our main result is an O(log k)-approximation algorithm for instances with k distinct job deadlines.

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each {\epsilon} > 0, any {\epsilon}-approximated formulation of P in the original space R^n requires a number of inequalities that is super-polynomial in n. This answers a question by Bienstock and McClosky (2012). We also prove that, for any down-monotone polytope, an {\epsilon}-approximated formulation in the original space can be obtained with inequalities using at most O(min{log(n/{\epsilon}),n}/{\epsilon}) different coefficients.

We study the {\em $k$-route} generalizations of various cut problems, the most general of which is \emph{$k$-route multicut} ($k$-MC) problem, wherein we have $r$ source-sink pairs and the goal is to delete a minimum-cost set of edges to reduce the edge-connectivity of every source-sink pair to below $k$. The $k$-route extensions of multiway cut ($k$-MWC), and the minimum $s$-$t$ cut problem ($k$-$(s,t)$-cut), are similarly defined. We present various approximation and hardness results for these $k$-route cut problems that improve the state-of-the-art for these problems in several cases. (i) For {\em $k$-route multiway cut}, we devise simple, but surprisingly effective, combinatorial algorithms that yield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. (ii) For {\em $k$-route multicut}, we design algorithms that improve upon the previous-best approximation factors by roughly an $O(\sqrt{\log r})$-factor, when $k=2$, and for general $k$ and unit costs and any fixed violation of the connectivity threshold $k$. The main technical innovation is the definition of a new, powerful \emph{region growing} lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a {\em different metric} for each source-sink pair. (iii) We complement these results by showing that the {\em $k$-route $s$-$t$ cut} problem is at least as hard to approximate as the {\em densest-$k$-subgraph} (DkS) problem on uniform hypergraphs.

In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph $G=(V,E)$, non-negative costs $c(v)$ and penalties $\pi(v)$ for each $v \in V$. The goal is to find a tree $T$ that minimizes the total cost of the vertices spanned by $T$ plus the total penalty of vertices not in $T$. This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving $O(\ln |V|)$-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally different primal-dual method achieving the same performance guarantee. Our algorithm introduces several novel features to the primal-dual method that may be of independent interest.

For a polytope P, the Chvatal closure P' is obtained by simultaneously strengthening all feasible inequalities cx <= b (with integral c) to cx <= floor(b). The number of iterations of this procedure that are needed until the integral hull of P is reached is called the Chvatal rank. If P is a subset of [0,1]^n, then it is known that O(n^2 log n) iterations always suffice (Eisenbrand and Schulz (1999)) and at least (1+1/e-o(1))n iterations are sometimes needed (Pokutta and Stauffer (2011)), leaving a huge gap between lower and upper bounds. We prove that there is a polytope contained in the 0/1 cube that has Chvatal rank Omega(n^2), closing the gap up to a logarithmic factor. In fact, even a superlinear lower bound was mentioned as an open problem by several authors. Our choice of P is the convex hull of a semi-random Knapsack polytope and a single fractional vertex. The main technical ingredient is linking the Chvatal rank to simultaneous Diophantine approximations w.r.t. the L1-norm of the normal vector defining P., Comment: 15 pages, 2 figures

We consider set covering problems where the underlying set system satisfies a particular replacement property w.r.t. a given partial order on the elements: Whenever a set is in the set system then a set stemming from it via the replacement of an element by a smaller element is also in the set system. Many variants of BIN PACKING that have appeared in the literature are such set covering problems with ordered replacement. We provide a rigorous account on the additive and multiplicative integrality gap and approximability of set covering with replacement. In particular we provide a polylogarithmic upper bound on the additive integrality gap that also yields a polynomial time additive approximation algorithm if the linear programming relaxation can be efficiently solved. We furthermore present an extensive list of covering problems that fall into our framework and consequently have polylogarithmic additive gaps as well.

In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known that it is NP-complete, if the number of colors is part of the input and that the problem is solvable in polynomial time, if the number of colors is at most 2. We also show that it is hard to satisfy almost all of the constraints for a feasible instance., Comment: minor revision

The symmetric Virtual Private Network Design (VPND) problem is concerned with buying capacity on links (edges) in a communication network such that certain traffic demands can be met. We investigate a natural generalization of VPND where the cost per unit of capacity may decrease if a larger amount of capacity is reserved (economies of scale principle). The growth of the cost of capacity is modelled by a non-decreasing concave function $f$. We call the problem the concave symmetric Virtual Private Network Design (cVPND) problem. After showing that a generalization of the so-called Pyramidal Routing problem and hence also the cVPND have the so-called tree routing property, we study approximation algorithms for cVPND. For general $f$, using known results on the so-called Single Source Buy at Bulk problem by Grandoni and Italiano, we give a randomized 24.92-approximation algorithm., Comment: Oberwolfach abstract

The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called the Pyramidal Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture. Here we consider a strengthened version of the PR Conjecture. First we establish several general tools which can be applied in arbitrary networks. Then we use them to prove that outerplanar networks satisfy the PR Conjecture.