We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the 1-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in ℝⁿ described by O(n) linear inequalities is hard to approximate within a factor of (1-ε)ln n for any constant ε > 0, unless P = NP. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

For an arrangement \(\mathcal{H}\) of hyperplanes in \(\mathbb{R}^n\) through the origin, a region is a connected subset of \(\mathbb{R}^n \setminus \mathcal{H}\). The graph of regions \(G(\mathcal{H})\) has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from \(\mathcal{H}\). We aim to compute a Hamiltonian path or cycle in the graph \(G(\mathcal{H})\), i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if \(\mathcal{H}\) is a supersolvable arrangement, then the graph of regions \(G(\mathcal{H})\) has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions \(\mathsf{P}(\mathcal{H}, R_0)\), obtained by orienting the graph \(G(\mathcal{H})\) away from a particular base region \(R_0\). Our second main result is that if \(\mathcal{H}\) is supersolvable and \(R_0\) is a canonical base region, then for any lattice congruence \(\equiv\) on \(\mathsf{P}(\mathcal{H}, R_0) =: L\), the cover graph of the quotient lattice \(L/{\equiv}\) has a Hamiltonian path.

Semialgebraic graphs are graphs whose vertices are points in {ℝ}^d, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on n vertices in this family, we can assign a label consisting of O(n^{1-2/(d+1) + {ε}}) bits to each vertex (where {ε} > 0 can be made arbitrarily small and the constant of proportionality depends on {ε} and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of O(n^{1/3 + {ε}}) bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of O(n^{1-1/d}log n) for d-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-d polynomial. Our labeling scheme is efficient in the sense that not only adjacency between two vertices can be decided in time linear in the size of their labels, but the labels can be computed in subquadratic time on a real RAM from the input points and the semialgebraic adjacency predicate, using recent polynomial partitioning algorithms. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size O(log n). We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size O(log³ n).

We consider the existence and construction of \textit{biclique covers} of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The \textit{size} of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size $O(n\polylog n)$. This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz's problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (\textit{Forum Math. Sigma}, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size $O(n\log^3 n)$. This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (\textit{Discrete Comput. Geom.} 1994). Finally, we prove that there exists families of unit disk graphs on $n$ vertices that do not admit biclique coverings of size $o(n^{4/3})$, showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider the existence of such cycles for a variety of polytopes, the facets of which have a natural combinatorial interpretation. In particular, we prove the following results:

An elimination tree for a connected graph \(G\) is a rooted tree on the vertices of \(G\) obtained by choosing a root \(x\) and recursing on the connected components of \(G-x\) to produce the subtrees of \(x\) . Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung–Hoang–Mütze–Williams combinatorial generation framework to elimination trees and prove that all elimination trees for a chordal graph \(G\) can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph \(G\) can be implemented in time \({\mathcal{O}}(\sigma)\) on average per generated elimination tree, where \(\sigma=\sigma(G)\) denotes the maximum number of edges of an induced star in \(G\) . If \(G\) is a tree, we improve this to a loopless algorithm running in time \({\mathcal{O}}(1)\) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of \(G\) , rather than just Hamilton path, if the graph \(G\) is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of \(G\) if and only if \(G\) is chordal.

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that $F(a_1,\ldots,a_k) = 0$. We consider a generalization of this problem in which $F$ is an arbitrary constant-degree polynomial, we are given $k$ sets of $n$ numbers, and have to determine whether there exist a $k$-tuple of numbers, one in each set, on which $F$ vanishes. We give the first improvement over the naïve $O^*(n^{k-1})$ algorithm for this problem (where the $O^*(\cdot)$ notation omits subpolynomial factors). We show that the problem can be solved in time $O^*\left( n^{k - 2 + \frac 4{k+2}}\right)$ for even $k$ and in time $O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right)$ for odd $k$ in the real RAM model of computation. We also prove that for $k=4$, the problem can be solved in time $O^*(n^{2.625})$ in the algebraic decision tree model, and for $k=5$ it can be solved in time $O^*(n^{3.56})$ in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for $k$-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft's point-line incidence detection problem in any dimension.

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs (SODA 2022). Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud (FPSAC 2022). This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group (SODA 2020). Our algorithms are derived from the Hartung-Hoang-Mutze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs. * Arturo Merino was supported by ANID Becas Chile 2019-72200522. Torsten Mütze was supported by Czech Science Foundation grant GA 22-15272S. Arturo Merino and Torsten Mütze were also supported by German Science Foundation grant 413902284.

We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle Δ∈C, the number of intersection points between the segments of A and those of B that lie in Δ. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+ε), for any ε>0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal et al. (2021) [3]. A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.

Abstract. In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. First, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage–Squire–West construction with a recent algorithm for generating elimination trees of chordal graphs. Second, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung–Hoang–Mütze–Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage–Squire–West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O (log log n )-competitive search tree data structure in this model, where n is the number of vertices. This matches the best-known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one that we call Steiner-closed search trees, which may be of independent interest. Moreover, our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees and, second, from these trees into paths.

An elimination tree for a connected graph G is a rooted tree on the vertices of G obtained by choosing a root x and recursing on the connected components of G–x to produce the subtrees of x. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-Mütze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph G can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph G can be implemented in time (m + n) per generated elimination tree, where m and n are the number of edges and vertices of G, respectively. If G is a tree, we improve this to a loopless algorithm running in time (1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of G, rather than just Hamilton path, if the graph G is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of G if and only if G is chordal.

We prove that some exact geometric pattern matching problems reduce in linear time to k -SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of $$k\ge 3$$ points within a set of n points in the plane, and for searching for an affine image of a set of $$k\ge d+2$$ points within a set of n points in d-space. As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height.

Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the order-oblivious setting. An algorithm A is instance-optimal in the order-oblivious setting for a certain class of algorithms 𝒜 if the following hold: - A takes as input a sequence of objects from some domain; - for any instance σ and any algorithm A' ∈ 𝒜, the runtime of A on σ is at most a constant factor removed from the runtime of A' on the worst possible permutation of σ. If we identify permutations of a sequence as representing the same instance, this essentially states that A is optimal on every possible input (and not only in the worst case). We design instance-optimal algorithms for the problem of reporting, given a bichromatic set of points in the plane S, all pairs consisting of points of different color which span an empty axis-aligned rectangle (or reporting all points which appear in such a pair). This problem has applications for training-set reduction in nearest-neighbour classifiers. It is also related to the problem consisting of finding the decision boundaries of a euclidean nearest-neighbour classifier, for which Bremner et al. (2005) gave an optimal output-sensitive algorithm. By showing the existence of an instance-optimal algorithm in the order-oblivious setting for this problem we push the methods of Afshani et al. closer to their limits by adapting and extending them to a setting which exhibits highly non-local features. Previous problems for which instance-optimal algorithms were proven to exist were based solely on local relationships between points in a set.

We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in $d$ dimensions, where $d$ is assumed to be a constant, in sublinear \textit{worst-case} update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with \textit{amortized} update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.

We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Δ\in C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Δ$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/\log^2n)\log^{O(1)}\log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+\varepsilon})$, for any $\varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the \emph{order type} of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.

Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n - 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n).

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O(log log n)-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.

Abstract Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $$\alpha$$ α -orientations of a graph G, in which every vertex v has a specified outdegree $$\alpha (v)$$ α ( v ) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $$\alpha$$ α -orientations of a planar graph G is at most two is -complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.

We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time O^*(eps^{-0.617}) where eps is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected Theta^*(eps^{-1}), and on all previous algorithms whenever eps = Omega(0.708^n). We also consider algorithms for 3-SAT with an eps fraction of satisfying assignments, and prove that it can be solved in O^*(eps^{-2.27}) deterministic time, and in O^*(eps^{-0.936}) randomized time. Finally, to further demonstrate the applicability of this framework, we also explore how similar techniques can be used for vertex cover problems.

In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any $$d>0$$ , the first algorithm maintains a proper $$O(\mathcal {C} dN ^{1/d})$$ -coloring while recoloring at most O(d) vertices per update, where $$\mathcal {C} $$ and $$N $$ are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an $$O(\mathcal {C} d)$$ -coloring with $$O(dN ^{1/d})$$ recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on $$N $$ vertices must recolor at least $$\varOmega (N ^\frac{2}{c(c-1)})$$ vertices per update, for any constant $$c \ge 2$$ .

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c

A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.

We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in ℝ3 can be colored with k colors so that every translate of the negative octant containing at least k6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer $k$, every finite set of points in $\mathbb{R}^3$ can be colored with $k$ colors so that every translate of the negative octant containing at least $k^6$ points contains at least one of each color. The best previously known bound was doubly exponential in $k$. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semionline model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set $P$, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of $P$. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical.

We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S by comparing a minimum number of pairs in T. Special cases include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing $ITLB+o(ITLB)+O(n)$ comparisons in the worst case. Here, n denotes the size of the ground sets, and $ITLB$ denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target partial order. Our approach is to replace the target partial order by a weak order (that is, a partial order with a layered structure) extending it, without increasing the information-theoretic lower bound too much. We then solve the problem by applying an efficient multiple selection algorithm. The overall complexity of our algorithm is polynomial. This answers a question of Yao [SIAM J. Comput., 18 (1989), pp. 679–689]. We base our analysis on the entropy of the target partial order, a quantity that can be efficiently computed and provides a good estimate of the information-theoretic lower bound.

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.

Proceedings of the 41st annual ACM Symposium on Theory of Computing STOC 2009, Bethesda, Maryland, 31 mai–2 juin 2009

When streaming packetized media data over a lossy packet network, it is desirable to use transmission strategies that minimize the expected distortion subject to a constraint on the expected transmission rate. Because the computation of such optimal strategies is usually an intractable problem, fast heuristic techniques are often used. We first show that when the graph that gives the decoding dependencies between the data packets is reducible to a tree, optimal transmission strategies can be efficiently computed with dynamic programming algorithms. The proposed algorithms are much faster than other exact algorithms developed for arbitrary dependency graphs. They are slower than previous heuristic techniques but can provide much better solutions. We also show how to apply our algorithms to find high-quality approximate solutions when the dependency graph is not tree reducible. To validate our approach, we run simulations for MPEG1 and H.264 video data. We first consider a simulated packet erasure channel. Then we implement a real video streaming system and provide experimental results for an Internet connection.

Finding adequate packet transmission strategies for media streaming systems is a challenging algorithmic task. Recently, we proposed an efficient dynamic programming algorithm for streams in which the dependencies between packets, such as those prescribed between video frames by video codecs, can be modeled with a tree. In this contribution, we propose a heuristic algorithm for arbitrary dependency graphs. This algorithm consists of first transforming the dependency graph into a tree by adding dependencies, and then applying the dynamic programming algorithm on the tree thus obtained. The algorithm is both simple and efficient, as shown by experimental results on video sequences

We consider the problem of rate-distortion optimized streaming of packetized multimedia data over a single quality-of-service network using feedback and retransmissions. For a single data unit, we prove that the problem is NP-hard and provide efficient branch and bound algorithms that are much faster than the previously best solution based on dynamic programming. For a group of interdependent data units, we show how to compute optimal solutions with branch and bound algorithms. The branch and bound algorithms for a group of data units are much slower than the current state of the art, a heuristic technique known as sensitivity adaptation. However, in many real-world situations, they provide a significantly better rate-distortion performance.

We propose a dynamic programming algorithm for finding optimal transmission policies for a single packet in rate-distortion optimized media streaming. The algorithm relies on an optimality assumption holding in particular when both the forward and round trip times have exponential distributions. In the other cases, we use the assumption as a heuristic principle. Simulations show that for realistic channel models, the algorithm provides optimal solutions and can be significantly faster than the previous fastest exact algorithm. The proposed algorithm can be used as a preprocessing step for streaming mutually dependent packets.

We consider the problem of rate-distortion optimal streaming of packetized media with sender-driven transmission over a single-QoS network using feedback and retransmissions. For a single data unit, we prove that the problem is NP-hard and provide efficient branch and bound algorithms that are in practice much faster than the best known solution. For a group of interdependent data units, we show how to compute optimal solutions with branch and bound algorithms. The branch and bound algorithms for a group of data units are slower than the current state of the art, the heuristic sensitivity adaptation algorithm, but provide a significantly better rate-distortion performance in many real-world situations.

This paper presents the study of the properties of graph colorings that minimize the quantity of color information with respect to a given probability distribution on the vertices. The minimum entropy of any coloring is the chromatic entropy. Applications of the chromatic entropy are found in coding with side information and digital image partition coding. We show that minimum entropy colorings are hard to compute even if a minimum cardinality coloring is given, the distribution is uniform, and the graph is planar. We also consider the minimum number of colors in a minimum entropy coloring, and show that this number can be arbitrarily larger than the chromatic number, even for restricted families of uniformly weighted graphs.

Two parties, Alice and Bob, wish to distill a binary secret key out of a list of correlated variables that they share after running a quantum key distribution (QKD) protocol based on continuous-spectrum quantum carriers. We present a novel construction that allows the legitimate parties to get equal bit strings out of correlated variables by using a classical channel, with as little leaked information as possible. This opens the way to securely correcting nonbinary key elements. In particular, the construction is refined to the case of Gaussian variables as it applies directly to recent continuous-variable protocols for QKD.

We consider the problem of generalized multiple description coding of image data, in which an image is encoded into M descriptions sent over M unreliable channels, and any description subset is useful in reconstructing an approximation of the original image. In a two-description scalar quantizer, a quantizer index is simply mapped by an injective mapping, defined by an assignment matrix, on a pair of indices. We propose a novel M-description coding method based on index assignment matrices for scalar quantizers. Instead of using a straightforward M-description generalization with hypercubes, we propose to encode a quantizer index in a multistage fashion, each stage doubling the number of descriptions. This method provides a much higher flexibility, allowing for instance different redundancy allocations for different loss profiles and simple repacketization.

We consider the problem of data compression with side information at the decoder and propose novel algorithms for quantization of the side information. These methods build quantizers that minimize the rate at which the source can be encoded with a constraint on the entropy of the quantized side information. We study relationships between our methods and those of context quantization used for image and video coding, and agglomerative information bottleneck in document classification. Our algorithms are shown to perform well on simple experiments and are applicable to a wide range of distributed multimedia compression problems.

Motivated by recent advances in quantum cryptography with continuous variables, we study the problem of extracting a shared digital secret key from two correlated real values. Alice has access to a real value, X/sub A/, and Bob to another value, X/sub B/, such that I(X/sub A/; X/sub B/)>0. They wish to convert their values into a shared secret digital information while leaking as little information as possible to Eve. We show how the problem can be decomposed into two subproblems known in other contexts. The first is the design of a quantizer that maximizes a mutual information criterion, the second is known as coding with side information.

We present a new multiple description source coding scheme based on tree-structured vector quantization (TSVQ). In this scheme, the codebook of each decoder is organized in a binary tree. The encoding is greedy and based on a sequence of binary decisions as in traditional TSVQ. Each binary decision of the encoder corresponds to adding information on one of the available channels and the encoding complexity can be shown to be proportional to the total bitrate. We describe the encoder structure for the two-channel case, and propose an entropy-constrained design algorithm based on marginal return analysis. Experimental results on a Gaussian source are presented for various design parameters and the generalization of the scheme to more than two channels is outlined.

Multiple description coding aims at transmitting two mutually refinable descriptions of a source on two different channels. Multiple description coding is successfully applied to transmission of multimedia signals on diversity systems or packet-switched networks. We present simple algorithms for the design of multiply descriptive tree-structured vector quantizers with linear encoding complexity. These algorithms allow for both the rates and distortion constraints in a natural way and are shown to yield quantizers that are competitive with full-search codes in terms of rate-distortion performance.

A new algorithm for fractal compression of greyscale images is presented. It uses some previous results allowing the compression process to be reduced to a nearest neighbors problem, and is essentially based on a geometrical partition of the image block feature space. Experimental comparisons with previously published methods show a significant improvement in speed with no quality loss.

Summary form only given. We propose two new tree-based search algorithms for vector quantizers using an additive weighted distance measure, such as ECVQ (entropy constrained vector quantization) (Chou et al., 1989). Both algorithms are based on a recursive space division technique, and use a bounding object at each node of the tree, in order to quickly eliminate subsets of the codebook during the search. The structure is more general than the k-d tree and the algorithm performs an optimal search similar to the one analyzed by Berchtold et al. (1997). We prove a theorem that defines the necessary and sufficient condition for any set of points to be a valid bounding object, i.e. to define a lossless pruning rule for the additive weighed Euclidean distance. The first algorithm presented uses rectangles as bounding objects, and the other uses spheres. We experimentally compare our approach with another recent one (Johnson et al., 1996), and show that the new algorithm using bounding rectangles performs significantly better for medium and high bitrate coding (>0.1 bits/sample) of a Gaussian process. This algorithm uses approximately 29 times less multiplications than a full codebook search at 1 bits/sample.

We present a new variable rate tree-structured vector quantizer (TSVQ) design algorithm, in which the complexity-distortion tradeoff is explicitly managed using a Lagrangian optimization approach. The algorithm is greedy and uses subvector distortion measures to lower the encoding complexity. We show that we can obtain low complexity encoders for the Gauss-Markov source with similar distortion to that observed on standard variable rate TSVQ.