The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

In the list homomorphism problem, the input consists of two graphs G and H, together with a list L(v) \subseteq V(H) for every vertex v \in V(G). The task is to find a homomorphism phi:V(G) -> V(H) respecting the lists, that is, we have that phi(v) \in L(v) for every v \in V(H) and if u and v are adjacent in G, then phi(u) and phi(v) are adjacent in H. If H is a fixed graph, then the problem is denoted LHom(H). We consider the reflexive version of the problem, where we assume that every vertex in H has a self-loop. If is known that reflexive LHom(H) is polynomial-time solvable if H is an interval graph and it is NP-complete otherwise [Feder and Hell, JCTB 1998]. We explore the complexity of the problem parameterized by the treewidth tw(G) of the input graph G. If a tree decomposition of G of width tw(G) is given in the input, then the problem can be solved in time |V(H)|^{tw(G)} n^{O(1)} by naive dynamic programming. Our main result completely reveals when and by exactly how much this naive algorithm can be improved. We introduce a simple combinatorial invariant i^*(H), which is based on the existence of decompositions and incomparable sets, and show that this number should appear as the base of the exponent in the best possible running time. Specifically, we prove for every fixed non-interval graph H that * If a tree decomposition of width tw(G) is given in the input, then the problem can be solved in time i^*(H)^{tw(G)} n^{O(1)}. * Assuming the Strong Exponential-Time Hypothesis (SETH), the probem cannot be solved in time (i^*(H)-epsilon)^{tw(G)} n^{O(1)} for any epsilon>0. Thus by matching upper and lower bounds, our result exactly characterizes for every fixed H the complexity of reflexive LHom(H) parameterized by treewidth.

The minimum unsatisfiability version of a constraint satisfaction problem (CSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size set of constraints whose deletion makes the instance satisfiable. For a finite set Gamma of constraints, we denote by CSP(Gamma) the restriction of the problem where each constraint is from Gamma. The polynomial-time solvability and the polynomial-time approximability of CSP(Gamma) were fully characterized by [Khanna et al. SICOMP 2000]. Here we study the fixed-parameter (FP-) approximability of the problem: given an instance and an integer k, one has to find a solution of size at most g(k) in time f(k)n^{O(1)} if a solution of size at most k exists. We especially focus on the case of constant-factor FP-approximability. Our main result classifies each finite constraint language Gamma into one of three classes: (1) CSP(Gamma) has a constant-factor FP-approximation; (2) CSP(Gamma) has a (constant-factor) FP-approximation if and only if Nearest Codeword has a (constant-factor) FP-approximation; (3) CSP(Gamma) has no FP-approximation, unless FPT=W[P]. We show that problems in the second class do not have constant-factor FP-approximations if both the Exponential-Time Hypothesis (ETH) and the Linear PCP Conjecture (LPC) hold. We also show that such an approximation would imply the existence of an FP-approximation for the k-Densest Subgraph problem with ratio 1-epsilon for any epsilon>0.

Symmetric Datalog, a fragment of the logic programming language Datalog, is conjectured to capture all constraint satisfaction problems (CSP) in L . Therefore developing tools that help us understand whether or not a CSP can be defined in symmetric Datalog is an important task. It is widely known that a CSP is definable in Datalog and linear Datalog if and only if that CSP has bounded treewidth and bounded pathwidth duality, respectively. In the case of symmetric Datalog, Bulatov, Krokhin and Larose ask for such a duality (2008, Vol. 5250 of LNCS , 93–124). We provide two such dualities, and give applications. In particular, we give a short and simple new proof of the result of Dalmau and Larose that ‘Maltsev + Datalog ⇒ symmetric Datalog’ (2008, IEEE Symposium on LICS , 297–306). In the second part of the article, we provide some evidence for the conjecture of Dalmau (2002, Proc. ICALP , 414–425) that every CSP in NL is definable in linear Datalog. Our results also show that a wide class of CSPs–CSPs which do not have bounded pathwidth duality (e.g. the P -complete H orn -3S at problem)–cannot be defined by any polynomial size family of monotone read-once non-deterministic branching programs.

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). Egri et al. (SODA 2014) augmented this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in log space or is hard for NL. A conjecture of Larose and Tesson from 2007 forecasts that when LHOM(H) is in log space, then in fact, it falls in a small subclass of log space, the set of problems expressible in symmetric Data log. The present work verifies the conjecture for LHOM(H) (and, indeed, for the wider class of conservative CSPs with binary constraints), and by so doing sharpens the aforementioned dichotomy. A combinatorial characterization of symmetric Data log provides the language in which the algorithmic ideas of the paper, quite different from the ones in Egri et al., are formalized.

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is hard for NL. Our algorithm operates by reducing the lists in a complex manner based on a novel decomposition of an auxiliary digraph, combined with repeated applications of Reingold's algorithm for undirected reachability (2005). We also prove an algebraic version of this dichotomy: the digraphs without a circular N are precisely those that admit a finite chain of conservative polymorphisms satisfying the Hagemann-Mitschke identities. This confirms a conjecture of Larose and Tesson (2007) for LHOM(H). Moreover, we show that the presence of a circular N can be decided in time polynomial in the size of H.

We completely classify the computational complexity of the list $\bH$-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph $\bH$ the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric Krom monotone SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for a number of constraint languages Gamma, the complement of the constraint satisfaction problem CSP(Gamma) can be expressed in symmetric Datalog. In particular, we show that if CSP(Gamma) is first-order definable and Lambda is a finite subset of the relational clone generated by Gamma then notCSP(Lambda) is definable in symmetric Datalog. Over the two-element domain and under standard complexity-theoretic assumptions, expressibility of notCSP(Gamma) in symmetric Datalog corresponds exactly to the class of CSPs computable in logarithmic space. Finally, we describe a fairly general subclass of implicational (or 0/1/all) constraints for which the complement of the corresponding CSP is also definable in symmetric Datalog. Our results provide preliminary evidence that symmetric Datalog may be a unifying explanation for families of CSPs lying in L.